The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications

Abstract

Abstract-Novel microwave imaging systems require flexible forward solvers capable of incorporating arbitrary boundary conditions and inhomogeneous background constitutive parameters. In this work we focus on the implementation of a time-harmonic Discontinuous Galerkin Method (DGM) forward solver with a number of features that aim to benefit tomographic microwave imaging algorithms: locally varying high-order polynomial field expansions, locally varying high-order representations of the complex constitutive parameters, and exact radiating boundary conditions. The DGM formulated directly from Maxwell's curl equations facilitates including both electric and magnetic contrast functions, the latter being important when considering quantitative imaging with magnetic contrast agents. To improve forward solver performance we formulate the DGM for time-harmonic electric and magnetic vector wave equations driven by both electric and magnetic sources. Sufficient implementation details are provided to permit existing DGM codes based on nodal expansions of Maxwell's curl equations to be converted to the wave equation formulations. Results are shown to validate the DGM forward solver framework for transverse magnetic problems that might typically be found in tomographic imaging systems, illustrating how high-order expansions of the constitutive parameters can be used to improve forward solver performance.

Enhanced Detection of Magnetic Nanoparticles Using a Novel Microwave Ferromagnetic Resonance Imaging System

“…as a forward solver to support distinct high-order expansions for the fields, contrasts and contrast sources, but in a full 3D vector-field implementation. Further details on both 2D and 3D DGM-CSI can be found in [38,[57][58][59]. Figure B.1: First stage scenario depicted for a multilayer synthetic phantom in air containing fat (yellow), fibroglandular tissue (blue, "Fibro" = fibroglandular), and a tumour (red).…”

“…To that end, there has been interest in applying contrast enhancement to the modality to bolster its cancer detection and monitoring capabilities, leading to a number of studies of biocompatible substances that alter the gigahertz-range complex electrical permittivity of a tissue of interest, when properly targeted [26,27]. Further details on DGM-CSI can be found in [57,58].…”

“…in performance has been demonstrated through the use of a tissue-dependent mapping process and the practice of cycling back to low-frequency reconstructions [40], and a brief overview of this process is provided in Section 3.3.2. Further details specific to DGM-CSI can be found in [57,58], but a summary of the relevant quantities and definitions involved in the CSI cost functional is included here, since it pertains to the discussion of data and domain error that follows.…”

“…Implementations of CSI based on finite-element method (FEM) forward solvers and more recently, the time-harmonic discontinuous Galerkin (DG) formulation of Maxwell's curl equations, have proven effective when modelling complicated imaging environments such as resonant chambers, and the use of FEM-CSI and DGM-CSI in many two-dimensional and three-dimensional4.2. Introduction87imaging scenarios, for dielectric and magnetic targets, has been extensively studied[48,[57][58][59]92].While conventional MWI continues to advance towards clinical application as a low-cost complementary imaging tool, a competitive need to improve the modality's limited spatial and contrast resolution has steered research towards microwave contrast media. Such contrast enhancement is a relatively new extension to the field, and since changes in relative permeability (µ r ) have been largely ignored for biomedical applications due to the intrinsically non-magnetic environment of the human body, most of the studies that have been carried out have focused primarily on agents that modify the relative permittivity (ε r ) of the targeted tissues[26,27,47].…”

Non-Iterative Eigenfunction-Based Inversion (Niei) Algorithm for 2d Helmholtz Equation

“…Then for the test case introduced in Section 4, a separation resolution study similar to the one reported in [33] is conducted. We compare the eigenfunction inversion results with the results of a Discontinuous Galerkin Method Contrast Source Inversion (DGM-CSI) algorithm [36]. In addition, for the same test case and three other test cases, results of varying the truncation number N are presented.…”