Dielectric shimming can address B₁⁺ inhomogeneity at high fields, with various applications in body imaging at 3T and neuroimaging at 7T. Obtaining a suitable design for the dielectric pads, however, remains a non-trivial task as many degrees of freedom need to be considered during the design, such as their geometry and composition. A recent study presented an inverse design method, by alternatingly updating the (unknown) material distribution to minimize the target field error, and subsequently solving for the associated fields by evaluating a forward problem.¹ The contrast source inversion (CSI) method has previously been shown to be a more efficient method in electromagnetic inversion, avoiding the need for evaluating a forward problem in each iteration.² The CSI method also allows for appropriate regularization, by incorporating constraints on the reconstructed permittivity. In this work, we present a procedure based on the CSI method to obtain the required pad design.

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 B₁⁺ 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 B₁⁺ 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 B₁⁺ 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 B₁⁺ 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

The results obtained in this study suggest that the CSI method is a very powerful candidate to solve the nonlinear optimization problem associated with the design of dielectric pads. Although the positivity and target permittivity constraints are implemented in a rather ad-hoc manner, the method converges very quickly and in a stable manner. Other means of constraining the permittivity can also be integrated into this approach.⁶ All schemes reach convergence within less than 10 seconds, which means that this approach is a suitable candidate for extension into 3D.We have presented an efficient CSI method for the inverse design of dielectric pads. This approach yields an automated procedure to provide the required pad geometry for a given ROI and permittivity, and can be extended to 3D in a similar manner.