A.3. Appendix Review of microwave breast imaging
Figure A.3.1. Microwave breast imaging is implemented by encircling the breast with antennas that can function as either a transmitter or a receiver. It is a multi–illumination approach, so the breast is successively illuminated from transmitter Tx with known incident electromagnetic fields (uinc). The resulting scattered (uscat) and transmitted fields are received by receivers Rx positioned at q on the breast’s periphery S and recorded by a measurement system. The process is repeated by successively illuminating the breast from different directions with different transmitters. The aim of microwave tomography is to reconstruct the dielectric profile ϵ(r) of the breast at points r within imaging domain D from the measurements.
Figure A.3.2. Microwave breast imaging procedure. A breast (represented by a forward model for a numerical study or measurements of a patient) is successively illuminated by incident fields from different directions. Microwave tomography is a model–based modality that extracts internal tissue information from the resulting scattered and transmitted fields to iteratively reconstruct an approximation of the actual spatial distribution of dielectric properties of tissues in the breast interior. Different tissue types are distinguished from each other by their characteristic dielectric properties.
Medical imaging with microwave tomography is a noninvasive approach for breast health monitoring to complement X–ray mammography. For a typical imaging scenario, a multi–illumination method is implemented by encircling the breast with antennas that act as both transmitters and receivers as shown in figure A3.1. The breast is sequentially illuminated with incident electromagnetic fields from a transmitter. The field distribution changes in response to dielectric property variations as the microwaves propagate through the breast so the resulting scattered and transmitted fields encode the spatial distribution of these dielectric properties. Receivers located along the periphery of the breast detect the scattered and transmitted fields after the microwaves have been transmitted through the breast. Information related to the breast’s profile is then extracted from the field measurements.
Microwave tomography is a model–based imaging modality that extracts internal tissue information from the field measurements to reconstruct an approximation of the actual spatial distribution of the dielectric properties over a reconstruction model consisting of discrete elements. With microwave tomography, bulk tissue characterization is the goal rather than more detailed depiction at the cellular level.
The dielectric properties of the breast tissues are represented by a complex permittivity where the real and imaginary components infer the ability of the tissue to store and absorb microwave energy, respectively [23]. The breast tissue types corresponding to skin, adipose (or fatty), transition, fibroglandular, and malignant tissues are characterized by their dielectric properties, which is supported by a number of large–scale studies [5–6]. Therefore, the complex permittivity profile that is reconstructed to form an image may be used to distinguish different tissue types. Estimating values of the dielectric properties of tissues over the model in order to reconstruct an image of the interior of the breast is achieved by solving an inverse scattering problem. The inverse problem is non–linear, so the model values are estimated iteratively using a process summarized in Figure A3.2.
Imaging is carried out in two steps. In the first step, patient data are collected using the procedure shown in Figure A3.1. For a numerical experiment, an electromagnetic forward model comprised of tissues with dielectric properties reported from large–scale studies [5–6] is constructed with the techniques described in [24–25]. The model is sequentially illuminated with numerical incident fields, and the calculated scattered and transmitted fields received by the numerical antenna are stored.
Once the patient data are collected, the reconstruction step using the inversion algorithm is carried out. This second step starts with a trial guess of the distribution. The electromagnetic model of the breast is initialized with this guess. An array of numerical antennas within a simulated measurement chamber that approximates the actual experimental system surrounds the breast and sequentially illuminates the breast with numerical incident fields. The resulting calculated scattered and transmitted fields received at the numerical antennas are recorded. A cost functional measures the discrepancy between the measured and calculated fields, and an inverse solver computes the optimal change in the parameter profile of the electromagnetic model necessary to reduce the discrepancy between these data. The trial solution is updated with these changes, and the forward solver recalculates the electric fields. The process continues in this iterative manner—updating and refining the reconstructed profile—until the calculated and measured fields match which, in turn, implies that the reconstructed profile matches the actual profile.
Various inverse solvers used have been proposed, including the finite element method contrast source inversion (FEM–CSI) [7,8,26], Gauss–Newton method, and conjugate gradient least squares (CGLS) algorithm [27], conjugate gradient method [28], a full–wave inversion method based on wavelet transform [29], wavelet expansion [30], the Distorted Born iterative method [31,32], and an inversion method based on an inexact Newton–type algorithm [33]. A significant challenge encountered when implementing these inverse solvers is that the inverse scattering problem, along with being non–linear, is severely ill–posed. This occurs due to the very large number of elements used by the reconstruction model to capture fine spatial features of the breast. Meanwhile, there are a very limited number of independent measurement data. Hence, the number of reconstruction elements (i.e., the dimension of the solution space) far exceeds the number of independent data resulting in non–unique solutions. An ill–posed inverse problem manifests as small perturbations of the measurement data leading to large errors in the reconstructions, and the convergence to false solutions that fit the data but differ significantly from the actual solution.
To alleviate the ill–posedness of the inverse problem, reconstruction techniques typically incorporate prior information into the objective function by using some form of regularization.