## 4 Discussion  
In this paper, we faced the problem of fitting the dielectric spectrum of a blood sample in order to estimate the parameters of the single-pole Debye model and the single-pole Cole-Cole model. In particular, we compared the performance of two different algorithms, LMA and VPA, in terms of accuracy and precision with respect to the starting points of the parameters.  
Compared to the results of the previous study [25], the performance of both algorithms improved significantly by introducing constraints on the parameters to be reconstructed. This prompted us to assess the algorithms in even worse cases, taking initial points further and further away from the solution of the optimization problem. In consideration of the shown results, we can say that, in the tests performed with the updated versions of the algorithms, LMA was more stable than VPA and, as before, was considerably faster. The slowness of VPA is due to the large number of measurement points considered (\(n=1000\)), which makes the computation of SVDs (Singular Value Decomposition) particularly time consuming.  
The results are promising and research will continue by evaluating algorithms and their variants in increasingly realistic scenarios, adding noise to synthetic data, and also considering dielectric spectra of other biological tissues.  
Conceptualization, R.S. and S.C.; methodology, R.D. and G.B.; software, R.D.; validation, R.D. and G.B.; formal analysis, R.S., R.D. and G.B.; investigation, R.D. and G.B.; resources, R.S. and S.C.; data curation, R.D.; writing--original draft preparation, R.D.; writing--review and editing, G.B. and R.D.; visualization, R.D. and G.B.; supervision, R.S. and G.B.; project administration, R.S. and S.C.; funding acquisition, R.S. and S.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by MIUR (Ministero dell'Istruzione dell'Universita e della Ricerca), Italy, as part of PRIN 2017 project \"Microwave Biosensors: Enhanced Non-Invasive Methodology for Blood Glucose Monitoring\" and by MUR (Ministero dell'Universita e della Ricerca), Italy, as part of \"Programma Operativo Nazionale Ricerca e Innovazione 2014-2020\" (PON \"Ricerca e Innovazione\" 2014-2020) project \"Non-invasive Electromagnetic Green Devices and Methods for Advanced Medical Diagnostics\".  
The data presented in this study are available upon request from the corresponding author.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & \(\varepsilon_{\infty}\) & \(\varepsilon_{\text{s}}\) & \(\tau\) (s) & \(\alpha\) & \(\sigma_{s}\) (S/m) & **RMSRE** \\ \hline \multirow{5}{*}{\(s\)} & & \(x_{\text{true}}\) & 2.31 & 73.3 & \(8.76\times 10^{-12}\) & 0.1 & 1.97 & - \\ \cline{2-9}  & & LMA & 2.31 & 73.3 & \(8.76\times 10^{-12}\) & 0.1 & 1.97 & \(1.6\times 10^{-24}\) \\  & & VPA & 4.12 & 79.7 & \(1.39\times 10^{-11}\) & 0.181 & 1.88 & 14 \\  & & LMA & 12.9 & 71.8 & \(1.08\times 10^{-11}\) & 0 & 2.01 & 1.07 \\  & & VPA & 12.9 & 71.8 & \(1.08\times 10^{-11}\) & 0 & 2.01 & 1.07 \\  & & LMA & \(1.81\times 10^{-11}\) & \(1.63\times 10^{-12}\) & \(3.27\times 10^{-24}\) & \(1.23\times 10^{-13}\) & \(5.58\times 10^{-14}\) & \(4.03\times 10^{-23}\) \\  & & VPA & 4.59 & 13.9 & \(1.14\times 10^{-11}\) & 0.168 & 0.18 & 28.6 \\  & & LMA & 0.000136 & \(2.32\times 10^{-5}\) & \(4.08\times 10^{-17}\) & 0 & \(3.89\times 10^{-7}\) & \(1.36\times 10^{-9}\) \\  & & VPA & \(6.15\times 10^{-5}\) & \(1.05\times 10^{-5}\) & \(1.85\times 10^{-17}\) & 0 & \(1.76\times 10^{-7}\) & \(9.66\times 10^{-10}\) \\ \hline \end{tabular}
\end{table}
Table 6: Statistical diagnostics on fitting results. Set C of the initial points and 250 mg/dL glucose concentrations were considered.