Electric Properties Tomography (EPT) of tissue is performed through electromagnetic analysis of complex $$$B_1^+$$$ maps^1,2,3. A key challenge is the calculation of first and second order spatial derivatives of phase maps near tissue boundaries. The inaccurate reconstruction is due to the invalidity of the electromagnetic reconstruction models^4,5,6 and the difficulties to calculate spatial derivatives on voxelized data.

In this study, we investigate the latter aspect by exploring the impact in EPT reconstruction of: different size of finite-difference kernels (K); k-space truncation on the approximated second order spatial derivatives ($$$\frac{\partial^2}{\partial r^2}$$$) of the B1+ phase. We demonstrate that there is a fundamental limitation to accurately calculate $$$\frac{\partial^2}{\partial r^2}$$$ and we explore alternative strategies with theoretically milder artifact levels.

Phase-only EPT conductivity reconstruction⁷, is closely related to the analysis of second order phase derivatives. This reconstruction (Fig. 1/A eq.1) is affected by two main numerical approximations coming from: truncated k-space data (image convolution with a kernel W) (Fig. 1/A eq.2); the use of finite-difference kernels K (Fig. 1/A eq.3). The concurrent action of W and K (Fig. 1/A eq.4) explains the notorious errors in σ_EPT(r) reconstruction.

Exact Phase Derivatives (EPD)⁸ can be instead calculated in the Fourier domain without making use of the convolution with K-kernels (Fig.1/B, eqs.5-6). However, as shown in Fig. 1/B eqs.7-8, noise in the high frequencies is amplified by (2πik)ⁿ. This could be mitigated by using appropriate truncation strategies. In Fig. 1/C, the effect of a Gaussian window (w_G) is compared to a standard boxcar (w_B).

We first explored by means of simulations the impact of different finite-difference K-kernel sizes and of the k-space truncation (W) on $$$\frac{\partial^2}{\partial r^2}$$$ for σ_EPT(r) reconstruction. Two different geometrical structures were investigated.

Secondly, we investigated the possibility to reconstruct conductivity maps by computing EPD directly in frequency domain (σ_EPD(r)), therefore independently from K-kernels. The impact of the truncating windows w_G and w_B was also explored. σ_EPT(r) maps were compared to σ_EPD (r) maps through simulations and measurements.

Measurements were performed on a phantom (Fig. 2)⁹ in a 3T MR scanner¹⁰. Simulations were performed in Matlab¹¹ using the same dimensions/properties of the phantom and a realistic $$$B_1^+$$$ phase map. Phase-only conductivity reconstruction was considered.

Figure 3/a-c shows that larger K-kernels sizes lead to more extended boundary errors, resulting into edges smoothing. The unavoidable truncation of k-space results in an additional artifact in image domain (Gibbs ringing) which corrupts σ_EPT(r) reconstruction (Fig. 3/b-d). In particular, as shown in the plots of Fig. 3 and by the coefficient of variation (CV(σ_EPT)=SD(σ_EPT)/mean(σ_EPT), for region 4 as an example), the ringing effect perturbation depends on the mutual position of different compartments. Comparison between truncated (Fig. 3/b-d) and non-truncated (Fig. 3/a-c) k-space reconstructions shows that the Gibbs ringing artifact is not negligible. This artifact can be mitigated if larger differentiation kernels are adopted (because of their smoothing effect) (Fig. 3/b-d). However, this is not a desirable solution since larger K-kernels sizes introduce larger boundary error extent.

The EPD approach, which is independent on differentiation kernels, was also explored (Fig. 1-4). Simulated and measured σ_EPD(r) maps (Fig. 4/b-c-f-g) seem to detect tissue boundaries better than σ_EPT(r) maps (Fig. 4/a-e). However, σ_EPD(r) maps result corrupted if a boxcar window is used to truncate the k-space (Fig. 4/b-f). Small perturbations in the K-space periphery are amplified by the multiplicative factors (2πik)ⁿ (Fig. 1, Eq. 7-8) and thus lead to high frequency fluctuations in image space. However, when a truncating Gaussian window is applied, these fluctuations are less pronounced (Fig. 4/c-g). As shown in (Fig. 1/c), the impact of high frequency perturbations is strongly attenuated. Gibbs ringing is reduced, although this effect will always be present at boundaries due to K-space truncation.

In addition to model invalidities, EPT reconstructions are also affected by artifacts related to the truncation of k-space, and the use of differentiation kernels. Our results show that the effect of k-space truncation is not negligible and it depends on structure geometry. In addition, the boundary error extent is larger for larger K-kernels sizes.

A new frequency-domain reconstruction technique, independent on differentiation kernel was explored. This technique is still affected by perturbations but, when combined with appropriate apodization windows, it might enhance the detection of structure boundaries. Although our analysis was limited to second order phase derivatives, our findings could also be extended to first and second derivatives of $$$B_1^+$$$ amplitude or to any other derivative-based reconstruction technique like Quantitative Susceptibility Mapping.