J Korean Soc Magn Reson Med. 2014 Dec;18(4):303313. English. Published online December 31, 2014. https://doi.org/10.13104/jksmrm.2014.18.4.303  
Copyright © 2014 Journal of the Korean Society of Magnetic Resonance in Medicine 
Jaewook Shin,^{1} Joonsung Lee,^{2} MinOh Kim,^{1} Narae Choi,^{1} Jin Keun Seo,^{3} and DongHyun Kim^{}^{1}  
^{1}Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea.  
^{2}SIRIC, Yonsei University, Seoul, Korea.  
^{3}Department of Computer Science & Engineering, Yonsei University, Seoul, Korea.  
^{}Corresponding author: DongHyun Kim, Ph.D. Department of Electrical and Electronic Engineering, Yonsei University, 134 Sinchondong, Seodamungu, Seoul 120749, Korea. Tel. 82221235874, Fax. 8223132879,  
Received August 19, 2014; Revised September 30, 2014; Accepted October 02, 2014.  
This is an Open Access article distributed under the terms of the Creative Commons Attribution NonCommercial License (http://creativecommons.org/licenses/by  
Abstract
 
Purpose
Invivo conductivity reconstruction using transmit field (B_{1}^{+}) information of MRI was proposed. We assessed the accuracy of conductivity reconstruction in the presence of statistical noise in complex B_{1} ^{+} map and provided a parametric model of the conductivitytonoise ratio value.
Materials and Methods
The B_{1}^{+} distribution was simulated for a cylindrical phantom model. By adding complex Gaussian noise to the simulated B_{1}^{+} map, quantitative conductivity estimation error was evaluated. The quantitative evaluation process was repeated over several different parameters such as Larmor frequency, object radius and SNR of B_{1}^{+} map. A parametric model for the conductivitytonoise ratio was developed according to these various parameters.
Results
According to the simulation results, conductivity estimation is more sensitive to statistical noise in B_{1}^{+} phase than to noise in B_{1}^{+} magnitude. The conductivity estimate of the object of interest does not depend on the external object surrounding it. The conductivitytonoise ratio is proportional to the signaltonoise ratio of the B_{1}^{+} map, Larmor frequency, the conductivity value itself and the number of averaged pixels. To estimate accurate conductivity value of the targeted tissue, SNR of B_{1}^{+} map and adequate filtering size have to be taken into account for conductivity reconstruction process. In addition, the simulation result was verified at 3T conventional MRI scanner.
Conclusion
Through all these relationships, quantitative conductivity estimation error due to statistical noise in B_{1}^{+} map is modeled. By using this model, further issues regarding filtering and reconstruction algorithms can be investigated for MREPT. 
Keywords: MREPT; Conductivity mapping; Noise analysis 
INTRODUCTION

Magnetic resonance electric property tomography (MREPT) was recently reintroduced to noninvasively image the distribution of electric property in human body at Larmor frequency (1, 2). Electric conductivity has been proposed to be used in diagnostic applications (3, 4, 5, 6) and for specific absorption rate (SAR) calculation (7). Over homogeneous regions, the relationship between the electric properties and magnetic fields can be simplified as a Helmholtz equation of the radiofrequency (RF) transmit field, B_{1}^{+}. In MREPT, the electrical properties distribution can be reconstructed from B_{1}^{+} field information by using the Helmholtz equation. The B_{1}^{+} magnitude and phase information is acquired separately using different sequences. For example, B_{1}^{+} magnitude mapping techniques such as actual flip angle imaging (AFI) (8), multiple TR B1/T1 mapping (MTM) (9), double angle method (DAM) (10) and BlochSiegert method (11) have been proposed and B_{1}^{+} phase map has been generally measured using half the phase acquired by a spin echo (2) or multiecho gradient echo sequence (12) under the assumption that the B_{1}^{+} transmit and receive phases are identical. In addition, conductivity reconstruction using only the B_{1}^{+} phase has been introduced (13) and widely used (14, 15), however it can provide accurate conductivity values under the assumption that the spatial variation of B_{1}^{+} magnitude is negligible. For all these applications and measurements, the conductivity value is needed to be estimated under an acceptable error level.
In MREPT reconstruction process, systemic and statistic error exists. First, systemic error occurs due to invalidity of the spatial homogeneity assumption and of the assumption of equality between B_{1}^{+} phase (θ_{1}^{+}) and B_{1}^{} phase (θ_{1}^{+}). When using the Helmholtz equation, boundary artifact is presented at tissue boundaries because of the invalidity of the spatial homogeneity assumption. In a previous study (16), rigorous mathematical error analysis about the boundary artifact was performed and numerical equation which takes into account the inhomogeneity was derived from timeharmonic Maxwell equation. Also, the half value of measured phase,
The second error occurs due to the statistical noise in the B_{1}^{+} map acquired. This statistic error is a fundamental limiting factor for application of quantitative conductivity mapping because the Laplacian operator, a secondary spatial derivative on the Euclidean space, used in the Helmholtz equation amplifies the statistical noise in the B_{1}^{+} map. To overcome these limitations, filtering, fitting and integral techniques are applied for conductivity imaging process (2, 13, 14). However, analysis about quantitative error due to noise in B_{1}^{+} map has not been performed prior to the application of these mentioned techniques. Therefore, in this study, quantitative conductivity estimation error due to the presence of statistical noise in the B_{1}^{+} map is analyzed and the factors that affect the variation of conductivity estimation inside tissue are examined. In addition, a numerical monomial form is suggested as approximate model for quantitative conductivity estimation error. The results of this study can be utilized for optimal filter design, comparison among reconstruction methods and examination of validity in clinical applications.
MATERIALS AND METHODS

Simulation model and Conductivity estimation method
Noiseless B_{1}^{+} field distribution of radio frequency (RF) transmit field at Larmor frequency was calculated using the Bessel boundary matching (BBM) method (18) which is a very fast electromagnetic (EM) simulation method for simple segmented models. The BBM method was implemented in MATLAB (The Mathworks, Inc., Natick, MA, USA). The numerical simulation phantom shown in Fig. 1 was an infinitely long cylinder placed at the isocenter of a 32 rod quadrature coil and the locations of the RF shield and RF coil were fixed at radius of 45 cm and 16.5 cm, respectively. The simulated phantom's radius, conductivity, permittivity and Larmor frequency was fixed to 5.5 cm, 1.0 S/m, 80ε_{0}, and 128 MHz, respectively, unless otherwise mentioned. The phantom was assumed to be surrounded by a different material having radius of 11 cm. The resolution of the generated B_{1}^{+} map was 1 mm by 1 mm.

Conductivity σ(r) was evaluated using the Helmholtz equation of Eq. 1 which can be induced from Maxwell's equation. The Laplacian operator was calculated using the surrounding four pixels in the discrete image domain. In addition, conductivity reconstruction using only the phase information was performed using Eq. 2 which is the 2D formation of B_{1}^{+} phase based method at a single point (13).
Afterwards, as in Eq. 3, conductivity estimation of a segmented region, Ω, was calculated by averaging the evaluated conductivity values excluding the boundaries, since the applicability of Eq. 1 and 2 are guaranteed only over homogeneous regions where area of the region is S_{Ω}.
Quantitative Conductivity Estimation Error Analysis
To investigate the relationship between quantitative conductivity estimation error and statistical noise in the B_{1}^{+} map, Gaussian noise was added to the B_{1}^{+} map prior to MREPT processing. In reality, it is necessary to separately consider the noise in the B_{1}^{+} magnitude and phase maps due to the separate acquisition of these maps. Statistical noise in B_{1}^{+} phase map shows Gaussian distribution for SNR >> 1 (19) but, statistical noise in B_{1}^{+} magnitude map shows diverse distribution depending on the B_{1}^{+} mapping method (20). However, in this simulation study, the noise in B_{1}^{+} magnitude map was assumed to be Gaussian distributed which seems reasonable since we focus on quantitative conductivity estimation error not to pattern of the error. Quantitative conductivity estimation error was defined in terms of the normalized root mean square error (NRMSE) obtainable from Eq. 4 with the real conductivity value σ_{real} evaluated using N trials of Monte Carlo simulations. However, in the case of B_{1}^{+} phase based conductivity reconstruction, reconstructed conductivity value inside homogeneous regions includes systemic error. So, to consider only the effect of statistical noise, the systemic error was disregarded in this case.
A conductivitytonoise ratio (σNR) is defined which is the real conductivity value divided by the standard deviation γ_{noise} of the statistical noise in the conductivity map (Eq. 5). In the case of a 1 mm radius object which is identical to the simulated image spatial resolution, since the homogeneous region excluding the boundary is just one pixel, the standard deviation γ_{noise} is equal to the root mean square error (RMSE). Then the σNR is equal to 1/NRMSE_{1 mm}, where NRMSE_{Rmm} represents NRMSE value over a target tissue of R millimeter radius. Although, the σNR can vary according to size and location of the target object, a previous study (13) showed the spatial dependency of conductivity evaluation error to be marginal. Here we can regard 1/NRMSE_{1 mm} as the general σNR over an object since the resolution of the simulated B_{1}^{+} map is 1 mm. Thus we refer to NRMSE_{1 mm} as the standard NRMSE.
(Simulation Variable A) Magnitude and phase noise
As referred in the introduction, SNR of B_{1}^{+} map is one of dominant factors for the accurate estimation of conductivity. In MR scanner, the B_{1}^{+} magnitude and phase are measured separately using different sequences, and thus the accuracy of B_{1}^{+} magnitude and the accuracy of B_{1}^{+} phase are different. In this work, the effect of complex noise in B_{1}^{+} map on conductivity estimation was investigated. Additionally, to investigate the dependency of conductivity estimation due to noise in separately acquired B_{1}^{+} magnitude and phase maps, the estimation error due to B_{1}^{+} magnitude noise or phase noise was calculated individually. To evaluate the accuracy of conductivity estimates with noisy B_{1}^{+} maps, we performed EM simulations to obtain noiseless B_{1}^{+} maps. Then, we considered four different cases of noisy B_{1}^{+} maps generated by adding Gaussian noises.
Noisy B_{1}^{+} complex  adding complex Gaussian noise to B_{1}^{+} complex.
Noisy B_{1}^{+} magnitude and noiseless B_{1}^{+} phase  adding real Gaussian noise to B_{1}^{+} magnitude.
Noiseless B_{1}^{+} magnitude and noisy B_{1}^{+} phase  argument of noisy B_{1}^{+} complex.
Noisy B_{1}^{+} magnitude and noisy B_{1}^{+} phase (fixed noise level)  combining noisy B_{1}^{+} magnitude (case 2) and noisy B_{1}^{+} phase (case 3) using Euler's formula
Specifically, for B_{1}^{+} complex and B_{1}^{+} magnitude, the NRMSE of conductivity estimates was evaluated by varying the SNR of B_{1}^{+}. The SNR of B_{1}^{+} complex and magnitude map can be defined as the average intensity of B_{1}^{+} inside the region of interest over the standard deviation of noise. However, for B_{1}^{+} phase, the above definition of SNR was not used directly since the phase itself is not absolute, but phase synchronization at an interval of 2π. The standard deviation of phase noise is a reciprocal of SNR in MR images (19). Thus, for B_{1}^{+} phase, the NRMSE of conductivity estimates was evaluated by varying the SNR of MR images.
In addition, to investigate the specific effect of statistical noise in B_{1}^{+} magnitude on conductivity estimation, the NRMSE of conductivity estimates was evaluated by varying the SNR of B_{1}^{+} magnitude according to conductivity value and B_{1}^{+} phase with fixed noise level.
(Simulation Variable B) Tissue characteristic & Larmor frequency
The effect of tissue related variables on conductivity estimation was examined. There are many variables associated with tissue, but, in this simulation, only three variables which cause change to the transmit field distribution were dealt with. In addition, the NRMSE value was calculated from conductivity estimates simulated from B_{1}^{+} maps using human tissue conductivity values previously reported from an exvivo study (21). These were done for different frequency values. The specific methods are mentioned below.
1. Surrounding tissue conductivity
The NRMSE from the tissue of interest (Ω), which was surrounded by another tissue (Ψ) at a radius of 11 cm (Fig. 1), was calculated. To examine the effect of the surrounding tissue on the reconstructed conductivity, the SNR level of the B_{1}^{+} map inside the tissue of interest was fixed to 100 and the conductivity of the surrounding tissue (σ_{out}) was increased.
2. Larmor frequency
Electric property of human tissues depends on frequency. Therefore, it will vary with the Larmor frequency of main magnetic field, f. Using electric property values from previous study (21) (Table 1), the NRMSE was calculated for seven different tissue types at three Larmor frequencies, 64 MHz, 128 MHz and 300 MHz. Fat, white matter (WM), gray matter (GM), prostate, cerebrospinal fluid (CSF), heart and liver were selected for the simulations. Here, these human tissues were modeled as cylindrical shapes with fixed radius of 5 mm (Fig. 1).

3. Tissue size
This simulation was performed using different sizes of simulation phantom whose radius was altered from 1 mm to 10 mm. Over a homogeneous conductivity region, the conductivity map obtained from the aforementioned MREPT process was averaged to estimate the conductivity value. The number of pixels being averaged (N_{A}) used in the conductivity estimation, is approximately proportional to the area of the phantom. Therefore, N_{A} equal 1, 5, 13, 49, 253 for phantom of radius 1 mm, 2 mm, 3 mm, 5 mm, 10 mm, respectively.
Parametric Modeling of the error in the conductivity estimate
The relationships between the error in the conductivity estimates and other imaging parameters were investigated. The imaging parameters considered here are N_{A}, SNR of B_{1}^{+},f, σ_{real} σ_{out}. For each imaging parameter, x, we modeled that the estimation error, NRMSE, is inversely proportional to a power p of x (Eq. 6).
To determine the power of the variable, p, the logarithm of the estimation error was fitted to a linear line of χ and p was determined from the slope of the fitted line. To confirm the fitting accuracy, the R^{2} (determinant coefficient) value was calculated.
Experimental study
To verify the simulation results, a cylindrical phantom (radius = 5.5 cm and height = 18 cm) was built and imaged. The phantom was filled with a mixture of 0.5% NaCl and 0.5% agar gel. The phantom conductivity measured with a conductivity meter (HI8733N, Hanna Instruments) was 0.82 S/m at 26℃. The measured conductivity value can be used to the conductivity value at 3T MRI because, in the case of saline water, the electric conductivity value is independent of frequency (22). To measure the B_{1}^{+} magnitude, double angle method (DAM) acquired with 60°  180° and 120°  180° flip angle acquisitions was used (10). For B_{1}^{+} phase measurement, the half value of the phase of a turbo spin echo (TSE) sequence was used. The imaging parameters were as follows: TR_{TSE}/TE_{TSE} = 800/20 ms, TR_{DAM}/TE_{DAM} = 2500/20 ms, FOV = 128 × 128 mm^{2}, resolution = 1.0 × 1.0 mm^{2}, slice thickness = 4 mm, 9 slices, and single channel quadrature transmit/receive coil was used. The SNR of B_{1}^{+} magnitude map was calculated using the law of error propagation similar to the approach in (9) and gave a value of 48. In the case of TSE for B_{1}^{+} phase measurement; acquisitions were repeated to gather 40 data sets, each 10 data sets were separately averaged to make the SNR vary from 80 to 220. All experiments were performed on a 3T Trio TIM system (Siemens Medical Solutions, Erlangen, Germany). The results were compared using the same size phantom with same SNR variation as the simulations.
RESULTS

The comparison of conductivity estimation due to three types of noisemagnitude, phase and complex noise typeis illustrated in Fig. 2 to evaluate which error component is most influential. It seems that the magnitude noise in B_{1}^{+} rarely affects to the conductivity estimation (less than 5% error) according to the greendot line in Fig. 2a. However, conductivity estimation is sensitive to complex and phase noise as shown in the bluedash and blacksolid line. Hence, it can be interpreted that the phase information is dominant for general conductivity reconstruction method which is in agreement with previous study (13).

When SNR varies from 10 to 40, the effect of complex noise which contains magnitude and phase noise is more influential than the effect of only phase noise (Fig. 2a). Thus, it can be interpreted that sensitivity of conductivity estimation to magnitude noise increases as SNR decreases in the B_{1}^{+} magnitude. In detail, Fig. 2b shows the NRMSE curves with fixed phase noise level converge to each minimum NRMSE values as conductivity changes. This convergence means that it is not necessary to increase magnitude SNR more than a specific conductivitydependent level. In this study, we defined the minimum required B_{1}^{+} magnitude level that makes the NRMSE to be within 3% difference with minimum NRMSE. This resulted in a value of 61.6, 39.8, 24.5 for conductivity values of 0.1, 0.3, 0.8 S/m. These values were further investigated for conductivity commonly found in in vivo at 3T and were summarized in Table 2. Note that these values are independent of phase noise level (Fig. 2b). Using the pth degree parametric model, the proportionality relationship between 1/NRMSE and SNR of complex B_{1}^{+} map (SNR_{B1+}) was fitted to first degree (p=1) as in Eq. 7 with 0.9559 R^{2} value.

As shown in Fig. 2a, the graphs of conductivity estimation errors due to B_{1}^{+} phase and complex noise show comparable NRMSE value when each SNR is higher than about 40. Therefore, SNR of the B_{1}^{+} map in Eq. 7 can be regarded as the SNR of the acquired image for B_{1}^{+} phase mapping.
The red dashed line in Fig. 2a shows conductivity estimation error when using phase based method. The error graph shows conductivity reconstruction using phase based method is less sensitive to statistical noise in B_{1}^{+} phase map than using both B_{1}^{+} magnitude and phase information. However, conductivity reconstruction using phase based method contains additional systemic error since it neglects variation of B_{1}^{+} magnitude. When phase based method was used for cylindrical object with the radius of 5.5 cm, conductivity value was 20% overestimated as an additional systemic error.
Fig. 3 shows the error in the conductivity estimates for seven types of tissues with the fixed radius of 5 mm for different Larmor frequencies. In general, the accuracy of conductivity estimation increases along Larmor frequency under fixed SNR. This phenomenon can be explained by the fluctuation of B_{1}^{+} magnitude and phase distribution (23, 24) which increases with Larmor frequency and makes the estimated conductivity value immune against amplified noise in the Laplacian operator. Also, the accuracy of conductivity estimates increases as tissue conductivity increases. Therefore, these two factors, Larmor frequency and conductivity value, can be represented to be directly proportional to 1/NRMSE value (Eq. 8).

The fitting of the above results gave a R^{2} value of 0.996 and 0.9999 for B_{0} and σ of tissue respectively. In calculating the fitting degree for Larmor frequency, the dependence of tissue conductivity on frequency was taken into account. As an example, if the conductivity is estimated using Eq. 1 at 128 MHz, the minimum radius of the homogeneous region required for NRMSE to be less than 10% given a single slice of B_{1}^{+} map with a spatial resolution of 1 mm and the SNR of 100, are about 128 mm (WM), 99 mm (Liver), 90 mm (GM), 73.8 (Heart), 64.5 mm (Prostate), 33.4 mm (CSF).
To assess the influence of the presence of conductivity material at the outer tissue, the results of evaluated NRMSE with various conductivity values at the outer tissue is seen in Fig. 4a. As seen, the conductivity of the outer tissue has no effect on conductivity estimation of the inner tissue for fixed SNR at the region of interest. As a result, the power of simulation variable σ_{out} can be estimated to 0 thus the term which related to σ_{out} can be removed.

Fig. 4b shows the relationship between the conductivity estimation error and radius of the tissue. As the radius of the estimation region increases, N_{A} increases. As a result, the conductivity estimation error gradually decreases. More specifically, the NRMSE ratio between 1 mm radius and the others can be found to be
The conductivity estimation process is analogous to a kind of filtering process since the homogeneous region with radius R_{mm} is averaged to estimate σ and NRMSE. Thus, 1/NRMSE_{Rmm} of estimated conductivity value can be the σNR after filtering.
In summary, quantitative conductivity estimation error due to statistical noise can be represented as in Eq. 10. In addition, the conductivitytonoiseratio averaged over an R_{mm} radius (σNR_{Rmm}) can be defined as in Eq. 11.
The unit for frequency (f) is in MHz and σ_{tissue} is in S/m. Here, the proportional constant K is added, which directly relates to the σNR and the parameters considered. In our simulation model, the value of K was approximately 640,000. However, this value will depend on many factors such as object shape and location, coil size, etc. The SNR_{B1+} can be influenced by the B_{0} value. Here, we assumed that the parameters SNR_{B1+} and B_{0} value are independent.
In MREPT, given a fixed Larmor frequency and the conductivity value of target tissue (σ_{tissue}), the dependent parameters are SNR_{B1+} and the number of being averaged pixels (N_{A}). Therefore, the principle of Eq. 11 can be utilized for conductivity mapping. As an example, if we consider a malignant liver tissue of 1 mm object whose conductivity value is 0.66 S/m (at 100 MHz, (3)) and a conductivity map with σNR = 2 is required, the SNR_{B1+} needed has to be approximately 19,000. In reality, to overcome the lack of SNR, a 22 mm radius homogeneous region is required to estimate the conductivity value with σNR_{Rmm} = 2 using a 100 SNR_{B1+}. Therefore, when the conductivity of target tissue, maximum available SNR_{B1+} and B_{0} field of the MRI scanner is fixed, the minimum size of detectable tissue using the MREPT process with an averaging filter is determined.
Finally, the principles obtained from the simulation results were compared to experiment study (Fig. 5). According to Table 2, the SNR of the acquired B_{1}^{+} magnitude map of 48 was sufficient to minimize NRMSE value for a phantom with conductivity value of 0.82 S/m. Thus, the NRMSE value was calculated according to the SNR of the image used for acquiring B_{1}^{+} phase map (Fig. 5a). The error graphs obtained from simulation and experiment show similar results. Note that the experiment results contain systemic error due to invalidity of the assumption of equality between θ_{1}^{+} and θ_{1}^{}. However, the effect of this systemic error was insignificant for conductivity estimation because of the spatially symmetric pattern of this error which vanishes by averaging process (17). Figure 5b shows the histogram of estimated conductivity values which shows similarity between the distribution of conductivity estimation value for simulation and experiment result.

DISCUSSION

Reconstruction of conductivity image using MREPT process suffers from systemic and statistic error. Especially, statistical noise in B_{1}^{+} map restricts the feasibility of MREPT. Hence, in this study, the relationship between quantitative conductivity estimation error and the parameters that affect the MREPT process was investigated and modeled as in Eq. 11. The systemic constant K can be different value according to the shape of tissue and transmit, receive coil, but the general tendency shown in this study holds.
For quantitative conductivity imaging, SNR_{B1+} is the most dominant factor because the Larmor frequency and conductivity of target tissue are almost uncontrollable factors. Thus, in the step of B_{1}^{+} map acquisition, sufficient SNR_{B1+} have to be guaranteed. Also, SNR of B_{1}^{+} magnitude and phase map have to be considered separately because the phase and magnitude of the RF transmit field, B_{1}^{+}, is generally acquired using different sequences. In the case of B_{1}^{+} magnitude map, the minimum required SNR of B_{1}^{+} magnitude for maximizing the SNR of conductivity map depends on the conductivity value according to Table 2. The minimum required SNR may change according to B_{1}^{+} magnitude mapping method due to the different distribution of noise in B_{1}^{+} magnitude map method however it will not vary significantly from the values in Table 2. Therefore, the acquisition time of B_{1}^{+} magnitude map should be fixed to a minimum to achieve the SNR required and the additional scan time can be used for achieving higher SNR of the acquired image for B_{1}^{+} phase mapping.
To overcome the sensitivity to statistical noise in B_{1}^{+} map, a filter was applied to the conductivity reconstruction process (2). In this study, the conductivity estimation process using the average of the reconstructed conductivity value inside a homogeneous region roughly shows the effect of filtering. From statistics, there is an
Higher Larmor frequency value also reduces conductivity reconstruction error because, as mentioned, the increase of Larmor frequency yields rapid spatial variation of B_{1}^{+} phase inside the object (24). However, in case of high B_{0} such as 7T, the assumption about the equality of θ_{1}^{+} and θ_{1}^{} is broken and this results in the increment of systemic error. Hence, in using B_{1}^{+} map acquired at high field MRI scanners, the detection and correction of systemic errors in conductivity map is required.
Through all these relationships, quantitative conductivity estimation error due to statistical noise in B_{1}^{+} map can be modeled as in Eq. 11. The equation can be used to find necessary SNR_{B1+} and filter size according to the required conductivity differences between, for example, normal and malignant tissue with a certain size. In addition, using this method, further issues regarding filtering and reconstruction algorithms can be investigated for MREPT.
Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012009903).
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