## 2 Materials and Methods
### Warelia MBI: Parametric Radar Image Formation  
The Wavelia MBI system operates using 18 antennae arranged in a circle in a horizontal plane outside a cylinder. With the patient lying in the prone position, one breast is submerged at a time into the cylinder, which is filled with a creamy transition liquid. The liquid has dielectric properties similar to the ones of the human skin within the microwave frequency spectrum, thus favoring the penetration of the electromagnetic waves in the breast. The device illuminates the breast using low-power electromagnetic waves in the frequency range [0.5-4] GHz. The probe array moves vertically below the examination table and illuminates the breast at regular intervals of 5 mm. Coronal sections of the breast, of a given thickness (10 mm) are generated using the MBI data at each vertical scan position of the probe array. Partially overlapping consecutive coronal breast sections, formed per azimuthal sector of illumination based on multi-static radar detection technology, are integrated to form a 3D MBI image of the dielectrically contrasted interior breast tissues. As specified in prior publications on Wavelia MBI [13; 15], the multi-static radar imaging algorithm, which is employed for MBI image formation, is the Time-Reversal Multiple Slgnal Classification (TR-MUSIC) algorithm, which was originally conceived for the detection of obscured radar targets in heavily cluttered environments [17]. The intensity of the TR-MUSIC images gets maximized in the imaging pixels where the MBI sensor array illumination vector is more orthogonal to the noise subspace; thus, the image intensity is indicative of the probability for the presence of a dielectrically contrasted scatterer on each pixel of the image. The noise subspace is estimated at each frequency, by means of decomposition and analysis of the Multi-Static Frequency Response Matrix (MFRM) of the imaging array. The illumination vector of the imaging array, at each pixel p of the imaging scene and each frequency f, is defined as:  
\[\mathbf{G}_{\mathrm{sect}}(\mathrm{p},\mathrm{f})=\left[ \begin{array}{c} \mathbf{g}_{0}\Big{(}\mathrm{P}_{\mathrm{TR}_{\mathrm{sect},\mathrm{I}}}, \mathrm{p},\mathrm{f}\Big{)}\end{array} \right. \begin{array}{c}\mathbf{g}_{0} \Big{(}\mathrm{P}_{\mathrm{TR}_{\mathrm{sect},\mathrm{I}}},\mathrm{p},\mathrm{ f}\Big{)}\end{array} \right]^{\mathrm{T}} \tag{1}\]  
with:  
\[\mathbf{g}_{0}\Big{(}\mathrm{P}_{\mathrm{TR}_{\mathrm{sect},\mathrm{I}}}, \mathrm{p},\mathrm{f}\Big{)}=\mathrm{j}\cdot\mathbf{H}_{0}^{(1)}\bigg{(}\frac{2 \mathrm{rf}}{\mathrm{c}_{0}}\cdot\bigg{(}\sqrt{\mathbf{e}_{\mathrm{r}, \mathrm{trans}}(\mathrm{f})}\cdot\hat{\mathrm{d}}_{\mathrm{OutOfBreast},\mathrm{ i},\mathrm{p}}+\sqrt{\mathbf{e}_{\mathrm{r},\mathrm{InBreast}}(\mathrm{f})}\cdot\hat{ \mathrm{d}}_{\mathrm{InBreast},\mathrm{i},\mathrm{p}}\bigg{)}\bigg{)} \tag{2}\]the assumed underlying ElectroMagnetic (EM) wave propagation model for the antenna element at position \(\mathrm{p_{TR_{succ,t}}}\), \(\mathrm{H}_{0}^{(1)}\) the Hankel function of 1 st kind and 0 th order, \(\mathrm{c_{0}}\) the speed of light in vacuum, \(\mathrm{e_{r,trans}}\)(f) the known permittivity of the transition liquid at the frequency f, \(\mathrm{\hat{d}_{OutOfBreast,i,p}}\) an estimate of the distance travelled by the EM wave in the transition liquid up to reaching the imaging pixel p, \(\mathrm{\hat{d}_{InBreast,i,p}}\) an estimate of the distance travelled by the EM wave within the breast up to reaching the pixel p, and:  
\[\mathrm{\hat{e}_{r,InBreast}}(\mathrm{f})=\left(\mathrm{pc_{fib}}\cdot\mathrm{ \hat{e}_{r,fibroidanalar}}(\mathrm{f})+(1-\mathrm{pc_{fib}})\cdot\mathrm{\hat{e }_{r,adipose}}(\mathrm{f})\right)\cdot 10^{-2} \tag{3}\]  
the average permittivity of the background healthy tissues of the breast, defined as a weighted average (weighting by pc_fib) of the adipose tissue and fibro-glandular tissue \"mean\" dielectric properties, as derived by [PERSON] et al. [18]. The breast external envelope is reconstructed at first using the Wavelia MBI scan data.