### INTRODUCTION

_{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 Bloch-Siegert 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 multi-echo 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.

_{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 time-harmonic Maxwell equation. Also, the half value of measured phase, $\frac{1}{2}{\mathrm{\theta}}_{{1}^{\mathrm{m}}}=\frac{1}{2}\left({\mathrm{\theta}}_{{1}^{+}}+{\mathrm{\theta}}_{{1}^{-}}\right)\approx {\mathrm{\theta}}_{{1}^{+}}$, is assumed as the B

_{1}

^{+}phase due to the assumption regarding the equality of phase. However, as the main field (B

_{0}) increases, assumption about the phase equality becomes weaker and reconstructed conductivity map becomes inhomogeneous even over homogeneous regions (17). In the case of B

_{1}

^{+}phase based conductivity reconstruction, an additional systemic error occurs inside the object due to the nonnegligible B

_{1}

^{+}magnitude variation.

_{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

_{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 iso-center 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.

_{1}

^{+}phase based method at a single point (13).

_{Ω}.

### Quantitative Conductivity Estimation Error Analysis

_{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.

_{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.

##### [5]

$\mathrm{\sigma}\mathrm{N}\mathrm{R}\equiv \frac{{\mathrm{\sigma}}_{\text{real}}}{{\mathrm{\gamma}}_{\text{noise}}}=\frac{{\mathrm{\sigma}}_{\text{real}}}{\sqrt{{\sum}_{\mathrm{\Omega}}\frac{{\left({\mathrm{\sigma}}_{\text{real}}-\mathrm{\sigma}\left(\mathrm{r}\right)\right)}^{2}}{{\mathrm{N}}_{\mathrm{\Omega}}}}}\cong \frac{{\mathrm{\sigma}}_{\text{real}}}{\sqrt{\sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{{\left({\mathrm{\sigma}}_{\text{real}}-{\stackrel{}{\mathrm{\sigma \u0302}}}_{\mathrm{i}}\right)}^{2}}{\mathrm{N}}}}=\frac{1}{{\text{NRMSE}}_{\text{1 mm}}}$### (Simulation Variable A) Magnitude and phase noise

_{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

_{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.

_{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

_{1}

^{+}maps using human tissue conductivity values previously reported from an ex-vivo study (21). These were done for different frequency values. The specific methods are mentioned below.

#### 1. Surrounding tissue conductivity

_{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

*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

_{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

_{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).

*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

_{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

_{1}

^{+}rarely affects to the conductivity estimation (less than 5% error) according to the green-dot line in Fig. 2a. However, conductivity estimation is sensitive to complex and phase noise as shown in the blue-dash and black-solid 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).

_{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 conductivity-dependent 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

*p*-th 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.

_{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.

_{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.

_{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).

*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).

_{out}can be estimated to 0 thus the term which related to σ

_{out}can be removed.

_{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 $\frac{{\text{NRMSE}}_{\text{2 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{3 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{5 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{10 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}\approx 14.14:7.25:2.83:1$. This ratio is inversely proportional to the N

_{A}to a degree of 2/3 (0.9992 of R

^{2}value, Fig. 4c. Since N

_{A}is almost proportional to square of the radius, the NRMSE for a R

_{mm}object is related to the standard NRMSE

_{1 mm}as the following: .

_{mm}is averaged to estimate σ and NRMSE. Thus, 1/NRMSE

_{Rmm}of estimated conductivity value can be the σNR after filtering.

_{mm}radius (σNR

_{Rmm}) can be defined as in Eq. 11.

*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.

_{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.

_{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

_{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.

*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.

_{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 $\sqrt{\mathrm{N}}$ SNR improvement when N number of pixels with independent random noise from a homogeneous region is averaged. However, in MREPT, there is an additional SNR improvement (N

^{2/3}in Eq. [9]) because the Laplacian operator can be regarded as an average over several pixels, i.e., the center and the surrounding pixels. Note that the minimum size of a detectable target tissue is regulated according to the filter size.

_{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.

_{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.