Magnetic resonance electric property tomography (MREPT) was recently re-introduced 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, B₁⁺. In MREPT, the electrical properties distribution can be reconstructed from B₁⁺ field information by using the Helmholtz equation. The B₁⁺ magnitude and phase information is acquired separately using different sequences. For example, B₁⁺ 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 B₁⁺ 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 B₁⁺ transmit and receive phases are identical. In addition, conductivity reconstruction using only the B₁⁺ 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 B₁⁺ 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 B₁⁺ phase (θ₁⁺) and B₁^- phase (θ₁⁺). 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 B₁⁺ phase due to the assumption regarding the equality of phase. However, as the main field (B₀) increases, assumption about the phase equality becomes weaker and reconstructed conductivity map becomes inhomogeneous even over homogeneous regions (17). In the case of B₁⁺ phase based conductivity reconstruction, an additional systemic error occurs inside the object due to the nonnegligible B₁⁺ magnitude variation.

The second error occurs due to the statistical noise in the B₁⁺ 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 B₁⁺ 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 B₁⁺ 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 B₁⁺ 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.

The comparison of conductivity estimation due to three types of noise-magnitude, phase and complex noise type-is illustrated in Fig. 2 to evaluate which error component is most influential. It seems that the magnitude noise in B₁⁺ 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).

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 B₁⁺ 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 B₁⁺ 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 B₁⁺ map (SNR_B1⁺) was fitted to first degree (p=1) as in Eq. 7 with 0.9559 R² value.

As shown in Fig. 2a, the graphs of conductivity estimation errors due to B₁⁺ phase and complex noise show comparable NRMSE value when each SNR is higher than about 40. Therefore, SNR of the B₁⁺ map in Eq. 7 can be regarded as the SNR of the acquired image for B₁⁺ 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 B₁⁺ phase map than using both B₁⁺ magnitude and phase information. However, conductivity reconstruction using phase based method contains additional systemic error since it neglects variation of B₁⁺ 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 B₁⁺ 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 R² value of 0.996 and 0.9999 for B₀ 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 B₁⁺ 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 $\frac{{\text{NRMSE}}_{\text{2 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{3 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{5 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}:\frac{{\text{NRMSE}}_{\text{10 mm}}}{{\text{NRMSE}}_{\text{1 mm}}}\approx 14.14:7.25:2.83:1$. This ratio is inversely proportional to the N_A to a degree of 2/3 (0.9992 of R² 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: .

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 conductivity-to-noise-ratio 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 B₀ value. Here, we assumed that the parameters SNR_B1⁺ and B₀ 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 B₀ 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 B₁⁺ 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 B₁⁺ 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 θ₁⁺ and θ₁^-. 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.

Reconstruction of conductivity image using MREPT process suffers from systemic and statistic error. Especially, statistical noise in B₁⁺ 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 B₁⁺ map acquisition, sufficient SNR_B1⁺ have to be guaranteed. Also, SNR of B₁⁺ magnitude and phase map have to be considered separately because the phase and magnitude of the RF transmit field, B₁⁺, is generally acquired using different sequences. In the case of B₁⁺ magnitude map, the minimum required SNR of B₁⁺ 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 B₁⁺ magnitude mapping method due to the different distribution of noise in B₁⁺ magnitude map method however it will not vary significantly from the values in Table 2. Therefore, the acquisition time of B₁⁺ 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 B₁⁺ phase mapping.

To overcome the sensitivity to statistical noise in B₁⁺ 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.

Higher Larmor frequency value also reduces conductivity reconstruction error because, as mentioned, the increase of Larmor frequency yields rapid spatial variation of B₁⁺ phase inside the object (24). However, in case of high B₀ such as 7T, the assumption about the equality of θ₁⁺ and θ₁^- is broken and this results in the increment of systemic error. Hence, in using B₁⁺ 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 B₁⁺ 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.