Inverse Design of Dielectric Pads based on Contrast Source Inversion

Wyger Brink^{1}, Jeroen van Gemert^{2}, Rob Remis^{2}, and Andrew Webb^{1}

The CSI method aims to minimize the target field error in the region-of-interest (ROI; $$$D_{\rm r}$$$) and the object error in the pad domain ($$$D_{\rm p}$$$). The target field error functional is defined as follows:

$$F_{\rm r} ({\bf w})=η_{\rm r} \parallel B_1^+({\bf w})-B_1^{+,{\rm target}} \parallel_{D_{\rm r}}^2$$

where $$$B_1^+({\bf w}) = B_1^{+,{\rm inc}} + {\bf G}_{\rm r}^+ {\bf w}$$$ is the *B*_{1}^{+} field generated by the contrast sources $$${\bf w}=\chi {\bf E}$$$ within the pad domain. The object error functional is written as

$$F_{\rm p} (\chi,{\bf w})=η_{\rm p} \parallel \chi {\bf E}^{\rm inc}-{\bf w}+\chi {\bf G}_{\rm p} {\bf w} \parallel_{D_{\rm p}}^2$$

which ensures that the field values satisfy Maxwell’s equations. $$${\bf G}_{\rm r}^+$$$ and $$${\bf G}_{\rm p}$$$ are the Green’s tensors relating electric current to *B*_{1}^{+} and electric field, respectively.^{1} The contrast sources* $$${\bf w}$$$* and contrast $$$\chi$$$ are alternatingly updated using a conjugate gradient scheme to minimize the combined error functional $$$F=F_{\rm r} +F_{\rm p}$$$.

The CSI method was constrained to reconstruct only positive permittivity values by forcing the negative real part and the imaginary part of to zero. In addition, the reconstructed permittivity distribution is forced to either zero or a target permittivity value by means of regularized thresholding.

The method was evaluated in a two-dimensional model corresponding to a neuroimaging scenario at 7T. The incident field was generated by a circular array of line sources surrounding a transverse slice of the heterogeneous head model ‘Duke’ (IT’IS Foundation, Zurich, Switzerland).^{3} The target field is defined as 1 μT within a manually drawn ROI, and a phase exchange method was applied on the target *B*_{1}^{+} field to retain the convex nature of the functional with respect to the contrast source quantity.^{4}

Fig. 1
illustrates the reconstructed permittivity when the CSI method is executed
without and with positivity constraints on the reconstructed permittivity. The
unconstrained CSI method results in both positive and negative values for the
optimized permittivity and conductivity distribution, which clearly does not
represent a distribution of passive material,but does illustrate the ability of
the method to arrive at the desired target field. When the solution space is constrained
to positive permittivity values only, a physically valid (ε_{r}>1) and
passive (σ>0) permittivity distribution is obtained, however at the cost of a
slightly reduced *B*_{1}^{+}
homogeneity within the ROI.

Fig. 2
illustrates the optimized permittivity distributions when the permittivity distribution
is constrained to have a fixed relative permittivity value of 100, 200 or 300. The
feasibility of such homogeneous permittivity distribution is much improved
compared to the variable distributions shown in Fig. 1. It is clear that when
the permittivity is higher, a smaller amount of material is required, which is
in line with previous studies.^{5}

1. Brink WM, Remis RF, Webb AG. A theoretical approach based on electromagnetic scattering for analysing dielectric shimming in high-field MRI. Magn Reson Med 2015.

2. van den Berg PM, Kleinman RE. A contrast source inversion method. Inverse Probl 1997;13:1607–1620.

3. Christ A, Kainz W, Hahn EG, et al. The Virtual Family—development of surface-based anatomical models of two adults and two children for dosimetric simulations. Phys Med Biol 2010;55:N23–38.

4. Setsompop K, Wald LL, Alagappan V, et al. Adalsteinsson E. Magnitude least squares optimization for parallel radio frequency excitation design demonstrated at 7 Tesla with eight channels. Magn Reson Med 2008;59:908–915.

5. Teeuwisse WM, Brink WM, Haines KN, et al. Simulations of high permittivity materials for 7 T neuroimaging and evaluation of a new barium titanate-based dielectric. Magn Reson Med 2012;67:912–918.

6. Abubakar A, van den Berg PM. The contrast source inversion method for location and shape reconstructions. Inverse Probl 2002;18:495–510.

Fig. 1.
Inverse design results without and with constraints. The target B1+ field was
defined as 1 μT within the ROI (white). When constraining the solution space to
positive permittivity values, a physically valid and passive permittivity
distribution is obtained, at the cost of a slightly reduced B1+ homogeneity.

Fig. 2. Inverse
design results for different target permittivities. It is clear that when the
permittivity is increased, a smaller amount of material is required, which is
in line with previous studies.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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