## 3 Results and Discussion
### Method Validation  
Validation of the proposed SI-HLLME protocol was conducted by constructing accuracy profiles. The theory behind this concept is described in the Supplementary Material. In brief, the dashed blue lines are representative of the \(\beta\)-expectation tolerance limits (95% probability level) and the red lines are representative of the relative bias. When former lines fall inside the black dotted lines that correspond to a 15% acceptance limit, the proposed bioanalytical method is considered to be valid [33]. Figure 3: Contour desirability plots for PIR and MEL determination using BBD. Figure 2: Three-dimensional response surface and contour plots for (**A**) PIR and (**B**) MEL determination using BBD. Calibration curves were prepared in triplicate (\(n=3\)) for each experiment series (k = 3) using mixture standards at the following concentrations (\(n=5\)): 0.1, 0.5, 1.0, 2.0 and 4.0 ug mL\({}^{-1}\). These standards were in three series made using pooled blank urine samples and by spiking them with the appropriate quantity of MEL and PIR and ISTD. Table 3 and Table S3 (Supplementary Materials) summarize the validation results for PIR and MEL, respectively. Six drug-free human urine samples (\(n=6\)) were used in equal volumes to prepare a pooled blank sample for the assessment of the method's selectivity. Figure 4 shows overlaid representative chromatograms of human urine samples spiked with ISTD only and ISTD and MEL and PIR, respectively. As can be observed, no interfering peaks were found within the region of elution of the target analytes and the ISTD, confirming the selectivity of the herein proposed SI-HLIME method. Subsequently, the linearity of the SI-HLIME methodology was studied through the selection of the response function that is the most appropriate. This study aimed to minimize the risk of undesired errors that might occur in the future. The following regression models were investigated: weighted (1/X) and weighted (1/X\({}^{2}\)) linear regression, and \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**Validation Criteria**} \\ \hline Response function (linear regression) & Slope & Intercept (\(\times 10^{3}\)) & \(r^{2}\) \\ \hline (\(k^{\text{a}}=3\); \(m=5\); \(n=3\)) (0.1–4.0 ug mL\({}^{-1}\)) & & & \\ Day 1 & 0.6971 & \(-\)0.0375 & 0.9998 \\ Day 2 & 0.7015 & \(-\)0.0367 & 0.9996 \\ Day 3 & 0.7066 & \(-\)0.0399 & 0.9995 \\ Precision (\(k=3\); \(n=3\)) & & & \\ \hline C (ug mL\({}^{-1}\)) & \(s_{\text{r}}\) (\%) b  & \(s_{\text{R}}\) (\%) c  & \\ \hline
0.1 & 2.5 & 2.6 & \\
0.5 & 2.9 & 4.4 & \\
1 & 2.8 & 3.9 & \\
2 & 2.0 & 2.8 & \\
4 & 2.8 & 2.6 & \\ Trueness (\(k=3\); \(n=3\)) & & & \\ \hline C (ug mL\({}^{-1}\)) & \multicolumn{2}{c}{Relative bias (\%)} \\ \hline
0.1 & \(-\)2.2 & \\
0.5 & \(-\)5.7 & \\
1 & \(-\)0.62 & \\
2 & \(+\) 1.2 & \\
4 & \(+\) 3.4 & \\ Accuracy (\(k=5\); \(n=3\)) & & \\ \hline C (\%) & Relative \(\beta\)-ETI (\%) & \\ \hline
0.1 & [\(-\)12.92, 8.61] & \\
0.5 & [\(-\)12.82, 1.40] & \\
1 & [\(-\)7.54, 6.30] & \\
2 & [\(-\)10.49, 12.96] & \\
4 & [\(-\)4.50, 11.16] & \\ Linearity (\(k=3\); \(n=3\); \(m=5\)) & & \\ (0.1–4.0 ug mL\({}^{-1}\)) & & \\ \hline Slope & 1.042 & \\ Intercept & \(-\)0.024 & \\ \(r^{2}\) & 0.9998 & \\ LOD (ug mL\({}^{-1}\)) & 0.02 & \\ LLOQ (ug mL\({}^{-1}\)) & 0.1 & \\ \hline \end{tabular}
\end{table} 
Table 3: Validation results for the determination of PIR in urine. simple unweighted. The respective models were constructed by plotting the peak area ratio of each analyte versus the ISTD peak area. The upper and the lower \(\beta\)-ETI for each regression model were calculated using the back-calculated concentrations, as well as the intermediate precision (\(\mathrm{s_{R}}\)%) and the mean relative bias. As illustrated in Figure 5, the unweighted linear regression models fitting profiles for both, resulted in results' scattering that for all levels fell within the acceptance limits.