Reijer L. Leijsen^{1}, Wyger M. Brink^{1}, Cornelis A.T. van den Berg^{2}, Andrew G. Webb^{1}, and Rob F. Remis^{3}

The goal of Contrast Source Inversion – Electrical PropertiesTomography is to iteratively reconstruct the conductivity and permittivity of tissue from $$$B_1^+$$$ field data. Up to now, this method has only been implemented in two dimensions. Here we demonstrate a proof-of-principle of a three-dimensional (3D) extension, thereby potentially turning CSI-EPT into a volumetric EPT technique. We present the basic 3D CSI-EPT equations, illustrate the performance of 3D CSI-EPT for realistic 3D body models, and show that accurate reconstructions are obtained even at tissue boundaries. The influence of the magnitude of the electric field on the reconstructions is also discussed.

In Electrical Properties Tomography (EPT) we aim to reconstruct the conductivity $$$\sigma$$$ and permittivity $$$\epsilon$$$ of different tissues from transmit $$$B_1^+$$$ data collected within a region of interest. These electrical properties allow quantification of the specific absorption rate^{1}, can aid in the discrimination of cancerous and benign tissue^{2} and can be used in hyperthermia treatment planning for the modeling of electromagnetic fields^{3}.

EPT methods based on the homogeneous Helmholtz equation^{1,4-9} show erroneous behavior at tissue transitions due to the assumed locally homogeneous media and are sensitive to noise since spatial differentiation operators act directly on the data. Contrast Source Inversion-Electrical Properties Tomography (CSI-EPT) is an iterative reconstruction method that circumvents these problems by making use of a global integral approach^{10}. To date, CSI-EPT has only been implemented in two dimensions, while here we present a fully 3D implementation which takes all the EM field components into account.

In CSI-EPT, the body that occupies the domain $$$\mathbb{D}^{\text{obj}}$$$ is described by the dielectric contrast function $$$\chi=\epsilon_\text{r}(\mathbf{x})-1-\text{j}\sigma(\mathbf{x})/(\omega\epsilon_0)$$$, where $$$\sigma(\mathbf{x})$$$ and $$$\epsilon_\text{r}(\mathbf{x})$$$ are the conductivity and permittivity profiles, respectively. Furthermore, the contrast source $$$\mathbf{w}=\chi \mathbf{E}$$$ is introduced, with $$$\mathbf{E}$$$ the electric field. In CSI-EPT, both the unknowns $$$\mathbf{w}$$$ and $$$\chi$$$ are reconstructed by minimizing the objective function

$$F(\mathbf{w},\chi)=\frac{\|\mathbf{r}\|_{\text{obj}}^2}{\|\chi\mathbf{E}^\text{inc}\|_{\text{obj}}^2}+\frac{\|\rho\|_{\text{obj}}^2}{\|B_1^+\|_{\text{obj}}^2},$$

where the data residual

$$\rho=B_1^{+;\text{sc}}-\mathcal{G}_{\text{data}}\{\mathbf{w}\}$$

describes the mismatch between measured and modeled data, while

$$\mathbf{r}=\chi\mathbf{E}^\text{inc}-\mathbf{w}+\chi\mathcal{G}_{\text{obj}}\{\mathbf{w}\}$$

is the object residual, which describes the discrepancy in satisfying Maxwell’s equations. In the above equations,

$$\mathcal{G}_{\text{data}}\{\mathbf{w}\}(\mathbf{x})=\frac{\omega}{c_0^2}\tilde{\nabla}\cdot\int_{\mathbf{x}'\in\mathbb{D}^{\text{obj}}}G(\mathbf{x}-\mathbf{x}')\mathbf{w}(\mathbf{x}')\,\text{d}V$$

is the data operator that maps contrast sources to field data, while

$$\mathcal{G}_{\text{obj}}\{\mathbf{w}\}(\mathbf{x})=\left(\frac{\omega^2}{c_0^2}+\nabla\nabla\cdot\right)\int_{\mathbf{x}'\in\mathbb{D}^\text{obj}}G(\mathbf{x}-\mathbf{x}')\mathbf{w}(\mathbf{x}')\,\text{d}V$$

is the object operator mapping contrast sources to electric fields. Furthermore, $$$\tilde{\nabla}=-\frac{1}{2}(\mathbf{i}_x+\text{j}\mathbf{i}_y)\partial_z+\frac{1}{2}(\partial_x+\text{j}\partial_y)\mathbf{i}_z$$$ and $$$G$$$ is the scalar Green’s function of air.

Given initial guesses $$$\chi^{[0]}$$$ and $$$\mathbf{w}^{[0]}$$$, CSI-EPT first fixes the contrast function and updates $$$\mathbf{w}$$$ using a conjugate-gradient type update formula and subsequently updates $$$\chi$$$, while keeping the contrast source fixed^{10}. This process is iteratively repeated until $$$F$$$ falls below an error tolerance, after which, at iteration $$$N$$$, the conductivity and permittivity profiles can be obtained as

$$\sigma=\text{Re}\left[\left(\chi^{[N]}+1\right)\text{j}\omega\epsilon_0\right]\quad\text{and}\quad\epsilon_\text{r}=\text{Re}\left[\left(\chi^{[N]}+1\right)\right].$$

We consider the head of the 3D Ella model from the IT'IS Foundation^{11} with an isotropic voxel size of 2.5 mm and simulate $$$B_1^+$$$ data from this region, generated by 16 $$$z$$$-directed dipoles each of length 30 cm, uniformly positioned on a circle around the head with a radius of 20 cm. The sources have a trapezoidal current distribution and operate in quadrature mode at 128 MHz (corresponding to 3T).

Fig. 1 shows the reconstruction of the head model, which has many small tissue interfaces and contains high conductivity and permittivity tissue. The 3D reconstruction after 100 iterations clearly shows that the center transversal slices are the most difficult to reconstruct. After 1000 iterations the tissue structure becomes visible, while after 10000 iterations we obtain a good overall agreement with the true model. Due to the increased complexity of the extension to 3D, the reconstruction of this fine-scaled head model takes about 20 hours with a Matlab implementation on a standard laptop. Computation times may be reduced by preconditioning techniques that reduce the number of CSI-EPT iterations.

To study the influence of the electric field on the reconstructions, we consider the same antenna setup as before, but placed around the abdominal region of Ella (see Fig. 2) and we compare reconstructions obtained in quadrature and equal phase mode (no phase difference between sources). Even though equal phase would result in a nulling of the magnetic field, the reconstructions clearly show that regions with a relatively low electric field are the most difficult to reconstruct (see Fig. 3). Complementary antenna settings or active or passive $$$B_1$$$ shimming techniques can be used to improve reconstructions in these regions.

To investigate what tissue profiles CSI-EPT is able to retrieve, we have assumed noiseless $$$B_1^+$$$ data. In practice, both the amplitude and phase (collected in separate measurements) contain noise. Noise effects can be suppressed, however, by adding a regularization term to the objective function^{10}. More critical is the presence of a receive phase component in the measured phase. Future work will therefore focus on developing a suitable signal model and objective function that takes this phase component into account.

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Contrast (first and second row), conductivity (third row) and permittivity (fourth row) magnitude maps of the head of the model Ella^{11}. The true contrast magnitude (first column), the reconstruction after 100 iterations (second column), after 1000 iterations (third column) and after 10000 iterations (fourth column). The gap visible in (b) is due to air in the nasal cavity that does not contribute to $$$B_1^+$$$.

Illustration of the antenna setup around the object of interest (abdominal region of Ella^{11} with isotropic voxel size of 2.5 mm and in total 366048 voxels). On the left a transversal view and on the right a coronal view through the center of the object. The rectangular box is the object domain $$$\mathbb{D}^\text{obj}$$$ which is reconstructed in each iteration. The dots and lines represent the dipoles. This dipole arrangement aims to imitate the integrated body coil of a 3T MR system, while it also allows for easy comparison with 2D. Note that 3D CSI-EPT is not restricted to this setup.

Coronal slice of the center of the object domain showing the correspondence between the magnitude of the electric field and the reconstruction results. The left and right column show the results with the sources operating in quadrature mode and in equal phase mode, respectively. The true contrast (a), the magnitudes after 10000 iterations of the reconstruction (b,c), the error (d,e) and the magnitude of the electric field strength (f,g). Even though the equal phase setting is not a realistic setting due to nulling of the magnetic field, it clearly shows the influence of the electric field on the reconstruction.