## 2 Methods  
A microwave imaging prototype can be thought of as a synthetic aperture radar system where the imaging volume is illuminated sequentially from antennas placed on a surface, \(\mathcal{A}\), and backscattered reflections are recorded from antennas placed on a surface, \(\mathcal{A}^{\prime}\). In the monostatic case, where reflections are recorded at the same transmitting locations in the frequency range, \(\Omega\), the Delay-and-Sum (DAS) [4; 6; 7; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58] beamformer can we written as (in the frequency domain):  
\[I(\mathbf{r})=\int\limits_{\Omega}\int\limits_{\mathcal{A}}S^{\prime}_{\mathbf{ s},\mathbf{s}}(\omega)\exp j\omega\tau_{\mathbf{s}}(\mathbf{r},\ \omega)d\mathbf{s}\ d\omega \tag{1}\]  
where \(\tau_{\mathbf{s}}(\mathbf{r},\ \omega)=\int_{C(\mathbf{s},\mathbf{r})}\frac{1}{c( \mathbf{r},\omega)}d\mathbf{s}\) is the propagation delay from the transmitting antenna to the point of interest, \(\mathbf{r}\), and back to the receiving antenna along the propagation path, \(C(\mathbf{s},\mathbf{r})\), for the frequency and spatial-dependent propagation speed, \(c(\mathbf{r},\omega)\). \(S_{\mathbf{s},\mathbf{s}}\) is the backscattered signals for the antenna at \(\mathbf{s}\). Many uncertainties affect propagation delay estimation:  
* the exact propagation path, \(C(\mathbf{s},\mathbf{r})\), is not known as the tissue composition of the imaging volume is unknown in a screening context;
* human breast tissues are dispersive, but the frequency-dependent propagation speed for each tissue is not known exactly, \(c(\omega)\);
* the propagation speed, \(c(\mathbf{r},\omega)\), along the propagation path is also unknown as this depends on the exact tissue composition of the imaging volume. In practice, certain simplifying assumptions are made in the imaging operator:  
* the propagation path is assumed as the straight-line path from the antenna, \(\mathbf{s}\), to the point of interest, \(\mathbf{r}\). This has been found to have a minimal impact on accuracy, at worst \(3\,\mathrm{mm}\)[35];
* the propagation speed is generally assumed to be defined at the centre frequency of the illumination pulse, \(\omega_{c}\);
* the propagation speed is assumed to not vary spatially in the entire imaging domain. Although a preliminary numerical study indicated that localisation accuracy could be improved by adapting the propagation delay depending on paths within the breast, this is not practical in realistic scenarios [11]. The above simplifying assumptions have been used in all microwave patient studies to date [1; 3; 4], and this work further investigates if the propagation speed can be adapted per patient using parameter search algorithms. As biological tissues are non-magnetic, the effective propagation speed (at the centre frequency of the illumination pulse) can be written as \(c(\mathbf{r})=\frac{c_{0}}{\sqrt{\epsilon_{r}}}\) where \(c_{0}\) is the speed of light and \(\epsilon_{r}^{\prime}\) is the effective average relative permittivity of the medium at the centre frequency of illumination. The effective average relative permittivity is a weighted average of the average relative permittivity of the entire breast. Thus, the propagation speed can be approximated as \(\tau_{\mathbf{s}}(\mathbf{r},\varepsilon_{r})=2\frac{\sqrt{\varepsilon_{r}^{ \prime}}\|\mathbf{r}-\mathbf{s}\|}{c_{0}}\) Therefore, Equation (1) can be represented as follows:  
\[I_{\varepsilon_{r}^{\prime}}(\mathbf{r})=\int\limits_{\Omega}\int\limits_{ \mathcal{A}}S_{\mathbf{s},\mathbf{s}}(\omega)\exp j\omega\tau_{\mathbf{s}}( \mathbf{r},\,\varepsilon_{r}^{\prime})d\mathbf{s}\;d\omega \tag{2}\]  
This paper investigates suitable fitness functions to use in parameter search algorithms to estimate the effective average dielectric properties, \(\varepsilon_{r}^{\prime}\), summarised in Figure 1:  
1.