Introduction

Electrical properties tomography (EPT) facilitates the non-invasive retrieval of electrical tissue properties such as permittivity and conductivity with conventional MRI [1,2,3]. As a result, EPT gained some interest in the past years as a potentially new biomarker in cancer diagnosis and treatment planning [4,5,6,7,8]. Tissue conductivity is mainly a function of water content and ion concentration and was shown to increase in case of pathology [4, 6,7,8].

Electrical properties (EPs) are related to the radio frequency (RF) field used for excitation via the Maxwell equations [2, 9]. This complex-valued RF excitation field, commonly called \({B}_{1}^{+}\), is assessed based on the measurable estimates of its magnitude (\(\left|{B}_{1}^{+}\right|\)) and phase (\({\varphi }^{+}=\angle {B}_{1}^{+}\)). To be exact, the latter cannot be measured directly but is often approximated by the so-called transceive phase, \({\varphi }^{\pm }\) [10, 11]. Often the EPs are obtained under the assumption that the EPs are constant across one tissue type. Maxwell’s equations can then be rearranged to yield the EPs via the homogeneous Helmholtz equation [3, 11, 12]. The reconstruction of the EPs ideally requires knowledge of the complex \({B}_{1}^{+}\) field (\(\left|{B}_{1}^{+}\right|{e}^{i{\varphi }^{+}}\)), but simplified relations exist that allow a separate estimation of the permittivity or conductivity alone based on the RF field’s magnitude or phase, respectively [3, 11]. Phase-based EPT, i.e. disregarding \(\left|{B}_{1}^{+}\right|\), however, usually results in an overestimation of the conductivity, especially at 3 T [3, 10, 11]. Even though a few MRI methods exist that yield simultaneous information of the \({B}_{1}^{+}\) magnitude and its phase [13,14,15], the components of the \({B}_{1}^{+}\) field are commonly obtained by two separate scans [10, 11, 16]. Most EPT studies focus on conductivity-only mapping using phase-based EPT [4, 7] accepting the resulting bias in favor of shorter scan times as the \({B}_{1}^{+}\) magnitude is not needed. For transceive phase measurements often a spin-echo sequence is used as it combines high signal with robustness against field inhomogeneities [12, 17, 18].

In this study, we show that a stimulated echo sequence with a double angle configuration (referred to as ‘DA-STE’ for the remainder of this article), first described by Akoka et al. [19], is very well suited for EPT and for conductivity mapping in specific. The suggested sequence generates both a spin and a stimulated echo, from which the simultaneous determination of the (transceive) \({B}_{1}^{+}\) phase and the \({B}_{1}^{+}\) magnitude is possible. The latter does not provide sufficient signal-to-noise for high-resolution permittivity mapping, but we will show that it is ideal for bias correction in conductivity mapping. Hence, no additional time is needed for full homogeneous Helmholtz-based EPT. The proposed method is directly compared to the double angle method with gradient echoes [20, 21] in phantom measurements. Furthermore, it is shown for in vivo measurements that DA-STE can deliver accurate, bias-corrected conductivity maps.

Methods

The double angle stimulated echo (DA-STE) sequence

The suggested double angle stimulated echo (DA-STE) sequence including the relevant gradient moments is shown in Fig. 1(a). An extended phase graph [22, 23] diagram is shown in Fig. 1(b) to visualize the spin and stimulated echo formation. Generally, three RF pulses will generate up to four spin echoes and one stimulated echo depending on distance and dephasing [22, 23]. If the refocusing flip angle is not exactly 180°, the magnetization dephased by the first gradient moment will partly be ‘stored’ as modulated longitudinal magnetization [22, 23]. Longitudinal magnetization stays unaffected by gradient moments but relaxes with the time constant T1. The third RF pulse flips the modulated longitudinal magnetization back into the transverse plane to form a stimulated echo [22, 23]. In order to ensure proper separation of the stimulated echo path from the spin echo path, the spin echo is dephased by twice the dephasing gradient moment from the first period [24]. This is emphasized in the extended phase graph schematic in Fig. 1(b).

Fig. 1
figure 1

a simplified sequence diagram of the suggested DA-STE variant. b Schematic extended phase graph of the sequence in (a)

Full size image

The signal intensities of the pure spin echo (\({S}_{se}\)) and the isolated stimulated echo (\({S}_{ste}\)) in a double angle configuration with an initial excitation pulse of 90° (as shown in Fig. 1(a)) are given as follows [19, 22,23,24,25]:

$$S_{se} = M_{0} \sin \left( {\text{B}_{1} \cdot \frac{\pi }{2}} \right)\sin^{2} \left( {\text{B}_{1} \cdot \alpha_{0} } \right)e^{{ - TE/T_{2} }}$$
(1)
$$S_{ste} = \frac{{M_{0} }}{2}\sin \left( {\text{B}_{1} \cdot \frac{\pi }{2}} \right)\sin \left( {\text{B}_{1} \cdot 2\alpha_{0} } \right)\sin \left( {\text{B}_{1} \cdot \alpha_{0} } \right)e^{{ - TE/T_{2} }} e^{{ - TM/T_{1} }}$$
(2)

In Equations [1] and [2], the magnitude \({M}_{0}\) is proportional to the proton density. In case of incomplete magnetization recovery, the steady state longitudinal magnetization before excitation can be used instead [26]. TM and TE are the mixing time and the echo time, respectively (see Fig. 1(a)). T1 is the longitudinal relaxation time and \({\alpha }_{0}\) refers to the nominal flip angle (as indicated in Fig. 1(a)). Conveniently, B1 refers to the relative \(\left|{B}_{1}^{+}\right|\) which is calculated by the ratio of the actual and nominal flip angle (\(\alpha /{\alpha }_{0}\)). When TM << T1, the cosine of the echo intensity ratio directly yields an estimate of the local flip angle \(\alpha\) [19, 27]:

$$\alpha = \text{B}_{1} \cdot \alpha_{0} \approx \cos^{ - 1} \left( {\frac{{S_{ste} }}{{S_{se} }}} \right)$$
(3)

Thus, the sequence provides an estimate of the local flip angle along with the phase from the pure spin echo SE. It is worth noting that Equation [3] is independent of the flip angle of the first excitation pulse as it cancels out in the signal ratio \({S}_{ste}/{S}_{se}\). Setting \(\alpha\) to 90° (as shown in Fig. 1(a)) proves to be beneficial for the signal-to-noise ratio (SNR) for the transceive phase estimation (using both \({S}_{se}\)) and for the estimation of the local B1 map (as can be verified with Monte Carlo simulations). Note that in case of a non-uniform slice profile, a bias is introduced to Equation [3] that needs to be accounted for to obtain the actual B1. Further details are given in section “Simulations”.

The uncertainty of various B1-mapping methods, such as DA-STE among others, has been investigated in-depth by Pohmann and Scheffler [26]. In the frame of its impact on EPT, the proposed method is in addition compared to the robust but lengthy double angle method using two gradient echo acquisitions [21] (GRE-DAM). The uncertainty was estimated by Monte Carlo simulations and analytically by error propagation according to Pohmann and Scheffler [26]. More detailed information is found in the Supplementary Information.

The uncertainty of the spin echo phase was estimated by the inverse of the SNR of the spin echo magnitude image [28]. Due to the averaging of two phase measurements with opposite read out gradient polarity, the uncertainty of the transceive phase is given by \({SD}_{{\varphi }^{\pm }}=\sqrt{2}\cdot {SN{R}_{Magn}}^{-1}\), where \(SN{R}_{Magn}\) is the SNR of one of the two magnitude images. \(SN{R}_{Magn}\) was obtained by the ratio of the average signal magnitude within the object and the standard deviation of the real part of the image in the background (air).

Electrical property reconstruction

Under the assumption that the local EPs vary slowly, the conductivity \(\sigma\) was reconstructed based on the homogeneous helmholtz (hH) equation [12, 29] yielding:

$$\sigma = \frac{1}{{\mu_{0} \omega }}{\text{Im}} \left\{ {\frac{{\nabla^{2} B_{1}^{ + } }}{{B_{1}^{ + } }}} \right\} = \frac{1}{{\mu_{0} \omega }}\left( {\nabla^{2} \varphi^{ + } + 2\frac{{\nabla \text{B}_{1} \cdot \nabla \varphi^{ + } }}{{\text{B}_{1} }}} \right)$$
(4)

where \({\mu }_{0}\) is the magnetic vacuum permeability and \(\omega\) is the Larmor frequency. The conductivity \(\sigma\) is mainly related to the leading term on the right side of Equation [4], which contains only the phase. To avoid additional scan time, the second term containing B1 is thus often neglected yielding the phase-based reconstruction formula [3, 11]:

$$\sigma \approx \frac{{\nabla^{2} \varphi^{ + } }}{{\mu_{0} \omega }}$$
(5)

The phase-based approximation can result in errors of 20% at locations of high B1 gradients in the brain at 3 T [11]. Therefore, for an accurate estimation of the tissue conductivity, the B1 information is needed (cf. Equation [4]). Because the transmit phase \({\varphi }^{+}\) is not directly accessible in MRI, it is approximated by half the measurable transceive phase \({\varphi }^{\pm }\) [10, 11]. The latter is the superposition of the transmit and the receive phase. These phases are assumed to be similar for symmetrical objects up to field strengths of 3 T if a quadrature birdcage coil-like configuration is used for transmission and reception. The full hH equation and the magnitude-based expression for the permittivity are given in the Supplementary Information (see Equations [S3] and [S4]).

The local Laplacian and gradients for each voxel were estimated by fitting a second-order polynomial in the voxel’s neighborhood [7]. To mitigate tissue boundary errors, fitting was only applied to voxels that have magnitude values within 25% of the center voxel [7, 12]. The Eps were subsequently smoothed with a tissue boundary preserving median filter [7, 12] using the same strategy. The window size for the Laplace estimation was set to 7 × 7 × 7 voxels and 13 × 13 × 13 voxels for the median filter. Skull stripping and was performed using the standard software package FSL (FMRIB Software Library v6.0, Oxford, United Kingdom) [30]. For segmentation, the TSE brain images were mapped to the standard MNI average brain atlas [31, 32] using FSL’s linear and non-linear registration tool [33,34,35]. The resulting white and gray matter probability masks [36] were registered back to the TSE images, which had the same position and resolution as the DA-STE images. Binary masks were then generated for probability values above 80% to estimate the average conductivity of white and gray matter.

Imaging experiments

All scans were performed at 3 T (Magnetom Prisma, Siemens Healthineers, Erlangen, Germany) using a dual-tuned 1H/23Na quadrature head coil for transmission and reception (Rapid Biomedical, Rimpar, Germany). Acquisitions were performed both in a phantom and in the brain of a healthy volunteer. In vivo human experiments were approved by the local ethics committee and informed consent was obtained from the volunteer beforehand. The saline phantom contained a salt concentration of 3 g/L, resulting in a theoretically estimated conductivity of 0.5 S/m at the Larmor frequency and at room temperature [37]. Relaxation times were reduced to tissue-comparable relaxation times of T1 ~ 930 ms and T2 ~ 75 ms by adding 0.125 mM MnCl2.

The double angle method using two single gradient echo acquisitions [21] (GRE-DAM) is used as a reference method for the B1 map due to its similarity to the proposed method and its robustness to motion. For GRE-DAM, an estimate of the local flip angle \(\alpha\) is obtained by \(\alpha ={\text{cos}}^{-1}\left({S}_{2}/\left(2{S}_{1}\right)\right)\). \({S}_{1}\) and \({S}_{2}\) represent the signals of the gradient echo measurements with the flip angles \({\alpha }_{1}\) and \({\alpha }_{2}=2{\alpha }_{1}\), respectively. A long TR is used to ensure that the B1 map is largely free from T1 effects. The flip angle of the first and the second acquisition was set to 90° and 180°, respectively, to increase the SNR of the resulting B1 map.

The gold standard of the transceive phase is the average of two spin echo phase acquisitions with opposite readout gradient polarity and was used to verify that the phase from the first echo of the suggested DA-STE sequence indeed corresponds to the conventional spin echo phase and thus to the transceive phase.

The sequence parameters of the two reference methods (SE, GRE-DAM) and the suggested approach (DA-STE) are compiled in Table 1 for the phantom measurements. For DA-STE, the gradient moment A in Fig. 1(a) was set to induce a dephasing of 4π per voxel (~ twice the readout gradient moment) and RF spoiling was enabled for each slice. For the brain measurement, the field-of-view (FOV) was 192 × 168 × 120 mm3.

Table 1 Sequence parameters for the GRE-DAM, the SE and the proposed DA-STE sequence. Different settings for separate experiments are given in brackets. GRE-DAM and the SE are the reference methods for the B1 map and the transceive phase, respectively. In the brain, the second GRE-DAM measurement used a TR of 7.86 s due to SAR restrictions
Full size table

For tissue-boundary preserving estimation of the Laplacian and median filtering, a T2-weighted image was acquired with a 90°–180° turbo spin echo (TSE) sequence provided by the manufacturer of the system. The TR was set to 10 s with a TE of 105 ms and a receiver bandwidth of 200 Hz/px. The voxel size was 2 × 2 × 2 mm3 and scanning lasted 2:44 min.

Simulations

All simulations and calculations were performed using MATLAB R2019a (The MathWorks, Natick, MA) unless stated otherwise.

For DA-STE, the \({S}_{se}\) and \({S}_{ste}\) signals were obtained using a 2D Bloch simulation for an ensemble of 361 × 361 magnetization vectors to simulate the effects within the xz-plane. The effects of the gradient moments in the frequency encoding (x) and slice selection (z) direction are thus accounted for individually. Signal simulation was performed using the parameter settings of Table 1. The impact of selective RF pulses was derived using the hard pulse approximation [38, 39].

Due to selective excitation, the flip angle estimated by Equation [3] must be corrected. To this end, \({S}_{se}\) and \({S}_{ste}\) signals of DA-STE were simulated for varying relative B1 values between 0.5 and 1.5 with a step size of 0.05. The resulting apparent flip angles (or rather B1) were calculated according to Equation [3] and correction was performed by interpolating the actual (input) B1 as a function of the apparent (output) B1, B1,apparent. For interpolation MATLAB’s interp1 function with the method ‘spline’ was used. Note that the correction was calculated for a fixed T1 and T2 (T1/T2 = 1 s/100 ms) which can result in a potential bias for deviating T1, such as for fluids. The same was done for the GRE-DAM for which T1 effects are smaller due to the longer TR.

Results

Simulations of the DA-STE signals, \({S}_{se}\) and \({S}_{ste}\), using the settings in Table 1 are used to find the relation between the resulting B1,apparent and the actual B1 according to Equation [3]. Simulations were performed for relaxation times similar to brain tissue (T1/T2 = 1 s/0.1 s) and fluid (T1/T2 = 4 s/2 s) (see Fig. 2(a)). It is evident that B1,apparent underestimates the actual B1 for most B1 values in the evaluated range. In Fig. 2(b), the difference between the ‘fluid’-like curve and the ‘tissue’-like curve is shown with respect to the apparent B1.

Fig. 2
figure 2

a Simulated results of the relationship between the apparent B1 due to the slice profile and the actual B1 using the proposed DA-STE sequence. The tissue data was obtained using relaxation times T1/T2 = 1 s/0.1 s. For Fluid, relaxation times were set to T1/T2 = 4 s/2 s. b Plot showing the difference in the actual B1 value (simulated) of fluid-like and tissue-like voxels with respect to the (interpolated) apparent B1

Full size image

For validation purposes, a direct voxel-wise comparison of the B1 maps obtained with the GRE-DAM (B1,REF) and the DA-STE (B1,DA-STE) method is shown in Fig. 3(a). Linear regression indicates a good fit of B1,DA-STE with B1,REF. The parameters in relative units (\(\alpha /{\alpha }_{0}\)) of the fitted function yield a slope of 0.991 and an offset of -0.004. The Bland–Altman plot in Fig. 3(b) illustrates the difference of the B1 maps against the mean B1 value. The latter is defined as the voxel-wise mean B1 value of the two methods, (B1,REF + B1,DA-STE)/2. The offset appears to be constant for different B1 values and thus indicates a global bias of 0.01 in relative units.

Fig. 3
figure 3

a Binned scatter plot showing the correlation between B1,DA-STE and B1,REF. B1,REF was obtained using the double angle method with two single gradient echo acquisitions (GRE-DAM). The orange line indicates the result of a linear least-squares analysis. b Bland–Altman plot of the two methods with lines marking the average difference and the 95% confidence intervals

Full size image

The DA-STE method was compared to the conventional spin echo for the transceive phase and to the GRE-DAM for the B1 map. Exemplary slices of the transceive phase for the reference and the DA-STE are shown in Fig. 4(a). The shown transceive phase is the average of two separate phase measurements of opposite read out gradient polarities. The SNR of the corresponding magnitude images was measured to be around 160 for both the reference spin echo and the spin echo image of DA-STE. The uncertainty of the transceive phase was thus calculated to be 0.009 rad for both methods.

Fig. 4
figure 4

a Exemplary slice of the transceive maps obtained by the proposed DA-STE and the reference, a spin echo acquisition together with the relative difference in the same slice. b B1 maps obtained by the proposed DA-STE and the GRE-DAM reference method together with the relative difference in the same slice

Full size image

The GRE-DAM uncertainty was estimated using Monte Carlo simulations and had an average of around 0.002 (\(\alpha /{\alpha }_{0}\)) within the of B1 value range of [0.6, 1.3] (\(\alpha /{\alpha }_{0}\)). This is approximately three times less than the proposed DA-STE, which had an uncertainty of 0.006 (\(\alpha /{\alpha }_{0}\)) in the same B1 value range (see Supplementary Information Figure S1).

Corresponding conductivity maps are presented in Fig. 5(a) and Fig. 5(b). The conductivities are shown once with the phase-based reconstruction (Equation [4], i.e., neglecting the B1 dependent term) in Fig. 5(a) and once with the full homogeneous Helmholtz equation (Equation [5], i.e., taking the B1 into account) in Fig. 5(b). The phased-based conductivity reconstruction exhibits a bowl shape for the homogeneous phantom [11]. Similar maps are obtained with the reference and the proposed method. Using the full homogeneous Helmholtz equation, the conductivity becomes more homogenous. For the reference method, the average conductivity over the whole phantom was found to be (0.51 ± 0.03) S/m. Due to outliers at the boundary, voxels within a width of 5 pixels from the phantom’s edge were excluded. For DA-STE, the average conductivity is also (0.51 ± 0.03) S/m. The average relative difference between the DA-STE and the reference conductivity was (0.03 ± 3.00) %. The corresponding permittivity maps reconstructed with the full homogeneous Helmholtz equation and the magnitude-based expression along with relative difference maps are shown in Supplementary Information Figure S4(a) and (b).

Fig. 5
figure 5

a Conductivity maps obtained by phase-based EPT for proposed DA-STE and the reference together with the relative difference in the same slice. b Conductivity maps obtained by full homogeneous Helmholtz EPT for the proposed DA-STE and the reference together with the relative difference in the same slice

Full size image

Transceive phase and B1 maps of the in vivo brain experiment are compiled in Fig. 6. The figure shows the transceive phase and the B1 map for three different slices. The uncertainty of the transceive phase was approximately 0.02 rad for both DA-STE and the reference method. The uncertainty of the reference was only 4% lower. Line profiles in the brain tissue are also shown on the right of Fig. 6(a). The B1 maps in the brain estimated with DA-STE and the GRE-DAM are shown for the same slices in Fig. 6(b). Based on Monte Carlo simulations, the average uncertainty of the DA-STE B1 is estimated to be approximately 0.014 (\(\alpha /{\alpha }_{0}\)) in the brain tissue. The GRE-DAM B1 uncertainty is around 0.003 (\(\alpha /{\alpha }_{0}\)). Like for the transceive phase, line profiles of both, the DA-STE B1 and the reference B1, are shown. The corresponding brain slices of the TSE image acquired for the tissue boundary preserving Laplacian estimation and median filtering is shown in Supplementary Information Figure S3.

Fig. 6
figure 6

Three exemplary axial slices of the brain of a healthy volunteer. a Transceive phase maps of DA-STE (top row) and the reference (bottom row). Line profiles at the location indicated by the black line are shown on the right. b B1 maps of DA-STE (top row) and the reference (bottom row). Line profiles at the location indicated by the black line are shown on the right

Full size image

Using the transceive phase and the B1 map, conductivity maps were reconstructed for the same axial slices as in Fig. 6. Figure 7(a) shows the conductivity calculation based on the phase-based reconstruction (cf. Equation [5]), the conductivity based on the full equation (cf. Equation [4]) and the difference map between the two. Apart from the CSF, the bowl shape of the phase becomes evident in the difference map. In Fig. 7(b), the full hH EPT conductivity based on the reference measurements is shown alongside a map showing the relative difference between the reference conductivity and the DA-STE conductivity. For DA-STE, an average conductivity of 0.44 ± 0.06 S/m was found in white matter and 0.63 ± 0.12 S/m in gray matter. For the conductivity obtained with the reference methods, the values are 0.45 ± 0.08 S/m for white matter and 0.67 ± 0.12 S/m for gray matter. The permittivity results in the brain are shown in Supplementary Information Figure S5.

Fig. 7
figure 7

Conductivity reconstructions for the same axial slices shown in Fig. 6: a phase-based conductivity (top row), full hH conductivity (middle row) and the difference between the two (bottom row). b the full hH conductivity of the reference measurements and the relative difference to the full hH DA-STE conductivity

Full size image

Discussion

In the presented work, we investigated a double angle stimulated echo (DA-STE) method for its use in EPT. Due to the known difficulties of permittivity mapping at 3 T [16], all EPT reconstruction focused on conductivity mapping with the obtained B1 map for bias correction according to the full homogeneous Helmholtz reconstruction (Equation [4]). As a consequence, conductivity quantification in the human brain generally improved and yielded values that are closer to expectations.

The proposed method features a spin echo and thus allows for the direct measurement of the transceive phase. The local B1 information is obtained by the generation of only one additional echo, a stimulated echo. DA-STE is thus more time-efficient than the recently proposed Carr–Purcell multi-spin echo method for complex \({B}_{1}^{+}\) estimation [15]. To accomplish a stimulated echo free from other (overlapping) echo contributions, the dephasing of the spin echo signal needs to be large enough to avoid convergence of different pathways (see Fig. 1(b)) [24]. As stated by Akoka et al. [19], the mixing time TM should be minimized to avoid increased T1 relaxation effects leading to deviations from Equation [3].

Even when small mixing times are chosen, large T1 variation in the subject can bias B1 estimation. Even though the steady state longitudinal magnetization cancels out in the ratio of the echo signals (cf. Equation [3]), simulations showed that a residual T1 sensitivity pertains for short TR. This is most likely caused by remnant modulated longitudinal magnetization [40]. For most relaxation times of tissues, the deviation is small and negligible as the phantom measurements show, especially for long TR.

A 3D DA-STE sequence would contain considerable dead time and lead to unpractical scan times. A 2D multi-slice acquisition scheme, thus, makes efficient use of this mandatory long TR. However, due to the slice-selective pulses and the resulting flip angle distribution across the slice, the apparent B1 needs to be corrected to find the true B1. As shown in Fig. 2, the correction of the B1 depends on the relaxation times, especially on T1. While assuming one fixed T1 and T2 similar to brain tissues leads to negligible effects for different brain tissue types, larger deviations from the true B1 are expected in tissues or regions with a long T1 such as in CSF. The size of this T1-related bias on the conductivity is shown in the Supplementary Information Figure S2. It can be seen, that even for large differences in relaxation times, a bias in the conductivity is expected only in areas with large B1 gradients, specifically at lower B1 values. To avoid this, an additional correction fit for CSF using respective relaxation times can be used, for example.

A direct comparison of the DA-STE B1 map with the B1 map from the GRE-DAM sequence revealed very good agreement between the methods. A slight offset between the two B1 maps of 0.01 (\(\alpha /{\alpha }_{0}\)) is revealed in the phantom by the Bland–Altman Plot in Fig. 3(b) and can be observed in the relative difference map in Fig. 4(b). The small offset, however, is negligible for conductivity calculation and conductivity reconstruction according to the full homogeneous Helmholtz equation (Equation [4]). Conductivity reconstructions yielded values that are very close to expectations (0.5 S/m) with both methods in the phantom. With the focus on conductivity mapping, DA-STE profits from a high SNR transceive phase measurement at the cost of local B1 precision. The B1 containing term in the conductivity reconstruction formula (Equation [4]) can be regarded as a correction term that corrects for deviations at the periphery leading to the well-known bowl shape for phase-only reconstructions. The phantom’s permittivity reconstruction distinctly reveals the “anti-bowl” shape when the magnitude-only reconstruction is used (cf. Supplementary Information Figure S4). Like the conductivity, the permittivity becomes more homogeneous when using the full hH EPT reconstruction. In the brain, permittivity mapping proves challenging at 3 T due to higher B1 uncertainty, resulting in a strongly inhomogeneous permittivity map [16]. Nevertheless, permittivity values are in a reasonable range, supporting the presented method (cf. Supplementary Information Figure S5).

With the presented method, an overall improvement of the conductivity values could be observed, in the phantom as well as in the brain. The tissue conductivities in the human brain reported in literature [41, 42] are 0.34 S/m for WM and 0.59 S/m for GM and 2.14 S/m for CSF. The reconstructed average conductivities in white and gray matter (WM: 0.44 ± 0.06 S/m; GM: 0.63 ± 0.12 S/m) aligns well with literature values. The accuracy and precision in CSF is generally challenging because of the use of large kernels, partial volume effects and relaxation times in addition to cardiac pulsation [43]. For DA-STE, the B1 accuracy in CSF is additionally lowered when the adjustment of the local B1 is not extended to large T1 values.

In clinical practice, generally multichannel receive coils are used. However, to comply with the EPT model assumptions, the measurements were carried out with a birdcage coil and thus could not benefit from parallel acquisition techniques such as GRAPPA [44]. For EPT, accuracy of the transceive phase is crucial and thus the use of a multichannel reception coil requires a coil combination technique that ensures correct reconstruction of the transceive phase, such as for example the Roemer/SENSE approach [45, 46]. In particular for phase-based EPT, i.e. neglecting the B1 curvature, a coil combination technique was proposed by Lee et al. [47]. In the case of optimal coil combination, the acquisition time of the proposed sequence can be significantly reduced by enabling parallel acquisition techniques, making it a fast and accurate conductivity mapping method.

Compared to other methods that simultaneously measure the B1 magnitude and phase, such as multi-echo actual flip angle imaging (AFI) [13] and the recently proposed Carr–Purcell (CP) spin echo sequence [15], DA-STE offers distinct advantages. AFI is known to be prone to artifacts in vivo that stem mainly from incomplete spoiling of fat magnetization and thus needs adjustment [48]. Being spin echo based, DA-STE is robust against chemical shifts and field inhomogeneities. The CP sequence extracts the local flip angle information from the signal modulation across multiple echoes. Therefore, a long echo train is needed that ultimately leads to longer scan times for the same resolution and number of slices. Additionally, in the CP sequence, the refocusing pulses are typically around 90°, resulting in a transceive phase uncertainty that is roughly doubled to that of the proposed DA-STE transceive phase.

Conclusion

A multi-slice dual spin echo and stimulated echo sequence offers rapid acquisition of the B1 and the transceive phase for B1-corrected conductivity mapping of the brain. In conclusion, the proposed method shows excellent prospects for bias-corrected conductivity mapping in the clinics.