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An Iterative Method for Electrical Properties Tomography Based on the Helmholtz Decomposition for the Electric Field
Naohiro Eda1, Motofumi Fushimi1, and Takaaki Nara1
1The University of Tokyo, Tokyo, Japan

Synopsis

This paper proposes a novel, integral-equation-based (IE-based) reconstruction method for magnetic resonance electrical properties tomography (MREPT). The proposed method can reconstruct the electric field and EPs simultaneously based on the Helmholtz decomposition of the electric field. In the proposed algorithm, we iteratively apply the projections onto the spaces in which the true electric field is contained. An advantage of the proposed method is that convergence is theoretically guaranteed in this iterative process. The efficacy of the proposed method is validated through numerical simulations.

Introduction

Electrical properties tomography (EPT) noninvasively estimates conductivity and permittivity distributions in the human body given a radio-frequency (RF) magnetic field (H+:=(Hx+iHy)/2)1. The previous EPT methods accounting for spatial variations of EPs can be categorized into two types: partial-differential-equation-based (PDE-based) methods and integral-equation-based (IE-based) methods. The convection-reaction magnetic resonance electrical properties tomography (cr-MREPT) method solves the PDE for the reciprocal of admittivity, 1/γ, which includes the Laplacian of H+ and hence is sensitive to noise. On the other hand, the EPT methods3,4,5,6 have been developed based on the IEs for the electromagnetic fields, which are insensitive to noise. However, convergence is not guaranteed in the iterative processes of methods3,4 for solving the scattering problem or our previous method5, which does not require a 'background' field. At ISMRM2020, we proposed a direct EPT method6 for solving the IE for E. However, the ill-posedness of the IE causes the error of EPs in this method. In this paper, we propose an IE-based method to reconstruct the electric field E and admittivity γ:=σ+iωϵ simultaneously using the Helmholtz decomposition of E, where σ and ϵ are the electrical conductivity and the permittivity, respectively, and ω/2π is the Larmor frequency. The proposed method is shown schematically in Figure 1. We iteratively reconstruct a true electric field E by repeating the projections onto the vector spaces that contain the true E. The field starts from the solenoidal part of E, Esol, which can be computed from H+ data and the boundary admittivity value. Once the algorithm converges, which is theoretically guaranteed, the true E and admittivity are obtained simultaneously.

Method

We start with the Helmholtz decomposition8 of a true electric field E in a bounded region of interest (ROI) given by E=Esol+Eirr, where Esol and Eirr are the solenoidal and irrotational parts of E, respectively, which are expressed as follows6: Esol(r)=Ω(n×E(r))×14π|rr|dS+Ω(×E(r))×14π|rr|dV,Eirr(r)=Ω(E(r))14π|rr|dV. Substituting Faraday's law, ×E=iωμH, and Ampere's law, ×H=γE, into Eq.(1) yields Esol(r)=Ω(n×1γ(×H))×14π|rr|dSiωμΩH×14π|rr|dV. Assuming that Hz=0 and |H||H+| and denoting :=(xiy)/2, Eq.(3) is further rewritten as Esol(r)=Ω(n×1γ[izH+zH+4iH+])×14π|rr|dSiωμΩ[H+iH+0]×14π|rr|dV, showing that Esol can be computed using the measured H+ and the EP values on the ROI boundary. Next, define two vector spaces C1:={f|f=EsolΩ(g(r))14π|rr|dVfor any vector fieldg},C2:={f|fis proportional to×H}. The true E is contained in C1 for g=E and is also contained in C2 from Ampere's law E=1γ×H. Hence in Figure 1, the problem is to recover the true E as an element contained in C1C2 from given Esol (also contained in C1 for g=0). For this problem we propose the use of a convex projection algorithm7: starting from Esol, by repeating the orthogonal projections onto C1 and C2 in turn, the vector field is updated and approaches the true E, as shown in Figure 1. To be precise, for fnC1 at step n, its projection onto C2 is given by gn=(fn(×H)|×H|2)×H. Then, for gnC2, its projection onto C1 is given by fn+1=EsolΩ(gn)14π|rr|dV. Setting the initial field as f0:=Esol, we update the vector fields using Eqs.(7) and (8) iteratively. The convergence of the algorithm is theoretically guaranteed because C1 and C2 are convex sets7. It is remarkable that when the algorithm converges to obtain f=E, it holds that f(×H)|×H|2=E(×H)|×H|2=1γ from Ampere's law. Therefore, when E is estimated, γ can be automatically determined.

We verified our method through numerical simulations. In the simulations, the H+ data was computed using FEM software (COMSOL Multiphysics, COMSOL Inc.). The following two situations were modeled: Model 1 was a simple model including three spheres with higher conductivity (σ=1.0 S/m) and lower relative permittivity (ϵr=50) as compared to the background material (σ=0.5 S/m, ϵr=80), as shown in Figure 2. Model 2 is a head model9. We compared the proposed method with our previous direct method6.

Results & Discussion

The proposed method outperformed the previous method with Model 1 and Model 2 (Figure 3 and Figure 5). This may be due to the improvement of the ill-posedness of the previous method by adding the constraint that E is proportional to ×H in the proposed method. Thus, in the proposed method, the errors of EPs may be only caused by the approximation of H by H+. The proposed method can reconstruct true EP maps using Eq.(3) with the true H of Model 1, whereas the previous method cannot due to the ill-posedness (Figure 4). It is observed that artifacts around the center are reduced if H can be used. The difference between the proposed method and the method proposed by us (#1434) is whether the formulation includes H+ only or H. In contrast to the method (#1434), the method in this paper has a characteristic that it is formulated using H and hence has the potential to reduce the artifact if components of H other than H+ such as H can be incorporated. Combining the method10 to compute H with the proposed method will be an important aspect in future studies.

Conclusion

In this paper, we proposed a novel, IE-based EPT method using the Helmholtz decomposition, which was verified numerically. If all of the components of H are available, the proposed method has the potential to suppress artifacts, which is the problem in the conventional methods.

Acknowledgements

The present study was financially supported by Canon Medical Systems Corporation.

References

1. Katscher U, van den Berg CAT. Electric properties tomography: Biochemical, physical and technical background, evaluation and clinical applications. NMR Biomed. 2017;30(8):1-15.

2. Hafalir FS, Oran OF., et al. Convection-reaction equation based magnetic resonance electrical properties tomography(cr-MREPT). IEEE Trans. Med. Imag. 2014;33(3):777-793.

3. Hong R, Li S. 3-D MRI-Based Electrical Properties Tomography Using the Volume Integral Equation Method. IEEE Trans. Microw. Theory Tech. 2017;65(12):4802-4811.

4. Leijsen RL., et al. 3-D contrast source inversion-electrical properties tomography. IEEE Trans. Med. Imag. 2018; 37(9):2080-2089.

5. Nara T, Furuichi T, Fushimi M. An explicit reconstruction method for magnetic resonance electrical property tomography base on the generalized Cauchy formula. Inverse Problems. 2017;33(10):105005.

6. Eda N, Fushimi M, Hasegawa K, Nara T. Noniterative Electrical Properties Tomography Reconstruction Method Based on Three-Dimensional Integral Equation for the Electric Field. ISMRM2020, #3191.

7. Youla, Dan C., and Heywood Webb. Image Restoration by the Method of Convex Projections: Part 1 Theory. IEEE Trans. Med. Imag, 1982;1(2): 81-94.

8. Zhou, X. L. On Helmholtz's theorem and its interpretations. J. ELECTROMAGNET. WAVE, 2007;21(4): 471-483.

9. Aubert-Broche B., et al. Twenty new digital brain phantoms for creation of validation image data bases. IEEE Trans. Med. Imag, 2006;25(11): 1410-1416.

10. T. Nara and M. Fushimi. Computation of H- from H+ and its application to regularization for Magnetic Resonance Electrical Properties Tomography (MREPT). ISMRM2020, #3186.

Figures

Illustration of the proposed method. A general vector field is decomposed into an irrotational part and a solenoidal part by Helmholtz decomposition. The solenoidal part of E, Esol, can be computed from known values. The blue and green lines are the vector spaces C1 and C2, respectively. The solid and dashed arrows are orthogonal projections onto C1 and C2, respectively. The proposed method estimates EC1C2 by repeating these projections and EP values are automatically determined from E.

(left) Simulation phantom of Model 1, including three spheres as the abnormal regions. (right) True EP maps of eight slices as a rectangular montage. The upper and lower rows show the results for conductivity and relative permittivity, respectively.

Reconstruction results of the proposed method (left column) and the previous method (right column) for Model 1 as a rectangular montage. The upper and lower rows show the results for conductivity and relative permittivity, respectively. Table shows the L2-errors of the reconstruction results. The proposed method resulted in a closer estimation of the conductivity and relative permittivity value as compared to the previous method.

Reconstruction results of the proposed method (left column) and the previous method (right column) for Model 1 with true H as a rectangular montage. The upper and lower rows show the results for conductivity and relative permittivity, respectively. Table shows the L2-errors of the reconstruction results. The proposed method can estimate the true EP maps, whereas the previous method cannot.

True conductivity maps (left column) and reconstruction results of the proposed method (center column) and the previous method (right column) for Model 2 as a rectangular montage. Table shows the L2-errors of the reconstruction results. The proposed method resulted in a more accurate estimated conductivity value as compared to the previous method.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)
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