Introduction

In MR-based Electrical Properties Tomography (EPT), Electrical Properties (EPs, conductivity σ and relative permittivity ε_r) are reconstructed from complex radiofrequency B₁⁺ fields.

Physics-based MR-EPT reconstructions suffer from noise and boundary errors leading to large variations in in-vivo EPs reconstructions^1,2. Deep learning methods are more robust, but suffer from generalization problems for input data not present in training (e.g. pathologies)³. These problems lead to low confidence in the reconstructed EPs maps, problematic for clinical translation⁴.

We present a physics-based method to compute complex B₁⁺ maps from given (reconstructed) EPs maps. The error between the measured and predicted, complex B₁⁺ maps, is here used as surrogate of the accuracy/quality of the EPs maps given as input. We demonstrate this for simulated and measured data.

Theory

The EPs are related to the complex B₁⁺ field ($$$B_1^+=|B_1^+|e^{i\phi^+}$$$ with |B₁⁺|=magnitude and ф⁺=transmit phase) by⁵:
$$-\nabla^2B_1^+=\omega^2\mu_0\varepsilon_cB_1^+-\bigg(\frac{\partial{B_1^+}}{\partial{x}}-i\frac{\partial{B_1^+}}{\partial{y}}\bigg)(g_x+ig_y)-\frac{\partial{B_1^+}}{\partial{z}}g_z$$

where ω=Larmor frequency, µ₀=vacuum permeability, $$$\varepsilon_c=\varepsilon_0\varepsilon_r-i\frac{\sigma}{\omega}$$$ (complex permittivity), and $$$g_{x,y,z}=\frac{\partial}{\partial{x,y,z}}\ln{(\varepsilon_c)}$$$. By splitting this equation in real/imaginary parts, and approximating the derivative operators with central finite difference schemes, the |B₁⁺| and ф⁺ in a voxel are expressed as function of |B₁⁺| and ф⁺ of neighboring voxels and their EPs⁶. Then, by imposing Dirichlet boundary conditions based on the measured B₁⁺ fields (Figure 1A), |B₁⁺| and ф⁺ are estimated using the iterative Jacobi method (Figure 1B)⁷.
The difference D_φ/D_B between the ground truth (measured) and the predicted B₁⁺ fields ($$$D_B=|\hat{B_1^+}|-|B_1^+|$$$; $$$D_\phi=\hat{\phi^+}-\phi^+$$$) can be used to evaluate the quality/accuracy of the input EPs (reconstructed with EPT methods).

D_φ/D_B does not immediately reflect local errors in the estimated EPs maps. However, given the relation between B₁⁺ fields and the EPs, the second order derivative of D_φ/D_B can be used as surrogate error map to localize errors in the input EPs maps:
$$L_B=S(\nabla^2(D_B))$$
$$L_\phi=S(\nabla^2(D_\phi))$$
where S is a smoothing (Gaussian) function. L_φ/L_B thus represents the discrepancy between the estimated and actual EPs, reflected in |B₁⁺| and ф⁺.

Methods

The method was tested on simulated B₁⁺ field data (Sim4Life, Zurich MedTech, Switzerland), using a validated quadrature birdcage coil setup⁸, and measured data on a cylindrical phantom.

Experiment 1: The B₁⁺ field was reconstructed using correct (ground-truth) EPs as input, on a spherical phantom with two compartments having different EPs in noiseless and noisy (SNR100) cases (setup: Figure 2). This was compared to the simulated, ground-truth B₁⁺ field to demonstrate the validity of the proposed method.

Experiment 2: We tested the impact of: 1) global under- and over-estimation of the input conductivity; 2) erroneous reconstruction of the inner compartment size; 3) presence of an anomaly in the input conductivity map.

Experiment 3: Using a brain model with tumor⁸, we tested the reconstruction error: 1) when correct EPs are used in input (Figure 4A); and 2) when the input EPs did not include the tumor (Figure 4B), simulating a case of erroneous MR-EPT reconstruction.

Experiment 4: We tested the method on MRI data (3 T, Philips, Ingenia) using a cylindrical phantom (see Figure 5 for sequence details and input EPs).

Results and Discussion

Figure 2 shows the accuracy of the method and its robustness to noisy B₁⁺ data (noise in D_φ/D_B only arises from noisy input B₁⁺). Higher differences arise at the boundary due to small model imperfections, which will be addressed in future work. The predictions take ~2 minutes for a volume of 256x256x80 voxels of which ~1 million are contained within the reconstructed geometry.

Figure 3 shows that, compared to the reconstruction using the correct EPs as input (A), the presence of a global offset (B) leads to a higher global D_φ. Changing the inner compartment size (C) shows a D_φ offset in the inner compartment. The presence of an anomaly (D) shows a higher D_φ around the anomaly. Overall, L_φ clearly reflects these variations in the input conductivity.

Predicting B₁⁺ in the brain results in higher D_φ/D_B compared to the spherical phantom (Figure 4A), mainly originating from the CSF boundaries. Figure 4B clearly shows a higher D_φ and L_φ when the tumor is not provided in input

Figure 5 shows the applicability of the methodology to measured data, with comparable errors as observed in simulations. The error maps suggest ground-truth EPs are slightly lower.

Future study will address the impact of using the transceive phase instead of the transmit phase.

Conclusion

We developed a physics-based method for B₁⁺ predictions from EPs maps reconstructed using MR-EPT. The difference between the measured and predicted |B₁⁺| and ф⁺ can be used as surrogate for the accuracy/quality of the EPs maps, adding confidence to in-vivo MR-EPT reconstructions, where ground truth is not available.

Acknowledgements

This work received funding from the Netherlands Organization for Scientific Research (NWO; VENI grant no. 18078), the Artificial Intelligence working group of the EWUU alliance; and the ISMRM research exchange program.

References

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