## 2 Results and Discussion
### Validation of the Proposed Methodology  
After the optimization section, we proceeded to the validation of the proposed methodology (BA\(\mu\)E(C18)-\(\mu\)LD/LVI-GC-MS(SIM)) using blank urine matrices. In a first approach, the analytical thresholds were determined through the LODs where values in between 0.20 and 1.56 \(\mu\)g L\({}^{-1}\), were achieved. The linearity range was assessed between 10.0 and 1000.0 \(\mu\)g L\({}^{-1}\) (nine concentration levels). This range was chosen once it included the therapeutic, toxic and lethal concentration values for TCAs [45]. For the calibration plots, we started by plotting a conventional linear regression, which showed good linearity (\(r^{2}\geq 0.9914\)), and according to the lack-of-fit test (at the confidence level 95%), it presented a good fit for the present study, i.e., the Fcalc was always below the Ftab. Nevertheless, by studying the data dispersion, the relative residues were too high (>15%), and it was clearly visible a tendency in the residue plots, which showed heterogeneous profiles. As a result, heteroscedasticity was observed for all TCA plots, which might be caused by the large range of the concentrations considered. For this reason, ordinary least-squares linear regression method was not fit for the attained data, since it results in large errors, in particular for the lower concentrations. Given the evidence of heteroscedasticity, a weighted linear regression method was adopted, once it is the simplest and most effective way to compensate the data dispersion observed, especially for the lower concentration levels. The quality of the fit of weighted regressions can be evaluated by calculating the sum of percentage relative error (%RE). The appropriate weighting factor can be calculated from the inverse of the variance. Nevertheless, this is impractical, once it requires several determinations for each calibration point and a new calibration plot must be done every time that the methodology is applied. Therefore, an empirical weight based on concentrations (variable x) and responses (variable y) were used. Six weighting factors were tested, 1/y1/2, 1/y, 1/y2, 1/x1/2, 1/x and 1/x2. The data achieved shows that a weighting factor of 1/y resulted in the lowest %RE across the whole range. The weighting factor should be used in the calculation of the regression equation parameters as described in previous works [45; 46; 47; 48]. As stated before, the LLOQ was defined as the lowest concentration value in which it is possible to quantify any analyte with precision and accuracy, i.e., it is the concentration in which the RSD and %RE are lower than 20%. Table 2 summarizes the validation parameters, namely the LODs, LLOQs, calibration equations and the \(r^{2}\) values achieved. As previously mentioned, accuracy and precision were calculated for inter and intra-day at four spiking levels, as summarized in Table 3. The intraday accuracy and precision values were between \(-12.0\) and \(14.2\%\), and ranging from 0.4 to 15.8%, respectively. The interday accuracy and precision ranged from \(-8.2\) to 20.0%, and between 2.1 and 19.9%, respectively. These data show that the proposed analytical approach presented suitable levels of accuracy and precision for trace level analysis of TCAs in urine matrices. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**TCAs**} & **LODs** & **LLOQs** & **Calibration Equations** & \(\mathbf{r^{2}}\) \\  & **(\(\mu\)g L\({}^{-1}\))** & **(\(\mu\)g L\({}^{-1}\))** & **Calibration Equations** & \(\mathbf{r^{2}}\) \\ \hline
**AMT** & & & y = 34.4780 x \(-\) 0.0174 & 0.9974 \\ \hline
**MIA** & & & y = 7.3701 x \(-\) 0.0026 & 0.9974 \\ \hline
**TRI** & & & y = 13.0250 x \(-\) 0.0033 & 0.9988 \\ \hline
**IMP** & & & y = 3.5938 x \(-\) 0.0027 & 0.9960 \\ \hline
**MIR** & 0.39 & & y = 11.2264 x \(-\) 0.0026 & 0.9982 \\ \hline
**DOT** & 1.56 & & y = 29.1992 x \(-\) 0.0094 & 0.9978 \\ \hline \hline \end{tabular}
\end{table}