## 2 Materials and Methods
### Numerical Simulations  
The generation of the synthetic complex relative permittivity of blood plasma relies on quadratic fits relative to glucose-dependent Cole-Cole parameters reported in [24]; in particular, we consider two different concentrations, 100 mg/dL and 250 mg/dL, where the former is a normal value while the latter is typical of severe diabetes, respectively, in accordance with the diagnostic criteria in [35]. The data vector consists of \(M=1000\) points in the frequency range 500 MHz-20 GHz. In gradient-like algorithms, the choice of the initial point is a crucial factor for the convergence of the procedure. For the single-pole model case, it is fairly easy to exploit the physical meaning of the parameters to infer an initial estimate. However, since the noise can invalidate the initial estimate, we propose to study the robustness of the two algorithms with respect to the initial point. To this end, a Monte Carlo analysis is performed, iteratively evaluating the deterministic algorithms using a set of \(N_{\mathrm{sim}}=1000\) uniformly distributed random initial points arranged in a 5D hypercube of the parameter space in order to statistically characterize the results. Each side of the hypercube represents an interval containing the range of variation of each parameter for the glucose concentrations considered. We run simulations on a machine with Intel i9-10850K (10 physical cores), 32 GB RAM, and Ubuntu 21.04, and we took advantage of the Parallel Computing Toolbox using the parfor loop for running the \(N_{\mathrm{sim}}\) simulations. The intervals for generating the random initial value for each parameter (of the Cole-Cole model) is chosen from the data tabulated in [24]. In particular, the widths of these intervals are the same for each glucose concentration. We define three sets of intervals for the starting points with wider and wider intervals. The first set, labeled as Set \(A\), consists of the following intervals: \([1,5]\) for \(\varepsilon_{\infty}\), \([1,150]\) for \(\varepsilon_{s_{\tau}}\)\([1\times 10^{-14},1\times 10^{-11}]\) for \(\tau\), \([0.1-1\times 10^{-9},0.1+1\times 10^{-9}]\) for \(\alpha\), and \([0,5]\) for \(\sigma_{\mathrm{s}}\). The second set, labeled as Set \(B\), consists of the following intervals: \([1,5]\) for \(\varepsilon_{\infty}\), \([1,150]\) for \(\varepsilon_{s_{\tau}}\)\([1\times 10^{-14},1\times 10^{-11}]\) for \(\tau\), \([1\times 10^{-4},0.6]\) for \(\alpha\), and \([0,5]\) for \(\sigma_{\mathrm{s}}\). The third set, labeled as Set \(C\), consists of the following intervals: \([0,10]\) for \(\varepsilon_{\infty}\), \([0,200]\) for \(\varepsilon_{s_{\tau}}\)\([1\times 10^{-14},1\times 10^{-10}]\) for \(\tau\), \([1\times 10^{-4},0.6]\) for \(\alpha\), and \([0,10]\) for \(\sigma_{\mathrm{s}}\). These intervals are relatively large compared to the values taken from [24] in order to test the two algorithms in sufficiently stressful situations. Obviously, it must be taken into account that VPA requires only the generation of \(\tau\) and \(\alpha\) values for the Cole-Cole model and only of \(\tau\) values for the Debye model. The algorithms are improved by introducing constraints on the parameters to be reconstructed. The required bounds are the following: \([0,30]\) for \(\varepsilon_{\infty},[0,200]\) for \(\varepsilon_{s},[1\times 10^{-14},\)\(1\times 10^{-10}]\) for \(\tau,[0,1]\) for \(\alpha,\) and \([0,10]\) for \(\sigma_{s}.\)  
For a quantitative evaluation of the performance of the two algorithms, we then define multiple figures of merit for statistically characterizing the results of the Monte Carlo analysis. For each parameter, mean and standard deviation are calculated over the entire set of reconstructions.