## 3 Results and Discussion
### Headspace Linearity and Its Impact on Analyte Relative Distribution
For MHS-SPME-TRIF experiments, slope (\(\The MHE approach is a dynamic, stepwise gas extraction carried out on the sample headspace; it was introduced for S-HS applications and adapted to HS-SPME, resulting in a technique referred to as MHS-SPME [44; 51; 52; 53; 54; 55]. It consists of three main steps:  
1. Exhaustive analyte extraction from calibration solutions in a range that matches real-sample concentrations. 2. Exhaustive analyte extraction from selected samples that show comparable matrix effects in order to define HS linearity boundaries. 3. Use of MHE on samples of interest. Steps 1 and 2 define the function according to the cumulative instrumental response that is the results of consecutive extractions from the same sample/calibration solution. After four to six consecutive extractions, in HS linearity [44; 53], exhaustiveness is accomplished and the decrease in the analyte response (chromatographic peak area) should be exponential. By summing the instrumental response (\(A\)) from each HS extraction, the total response that is virtually generated by the analyte amount in the sample (\(A_{T}\)) can be estimated--Equation (2):  
\[A_{T}=\sum\
olimits_{i=1}^{=\infty}A_{i}=A_{1}\;\frac{1}{(1-e^{-q})}=\frac{A_ {1}}{(1-\beta)}, \tag{2}\]  
where \(A_{T}\) is the total estimated response (chromatographic area), \(A_{1}\) is the response after the first extraction, and \(q\) is a constant associated with the response exponential decay (\(\beta\)) over consecutive extractions. The term \(q\) is obtained from the natural logarithm of the analyte response as a function of the number of extractions, and a linear regression equation (Equation (3)) can be calculated:  
\[\ln A_{i}=a\;(i-1)+b, \tag{3}\]  
where \(i\) is the number of extraction steps, \(b\) is the intercept on the y axis, and \(a\) is the slope. \(\beta\) (\(e^{-q}\)) is analyte and matrix dependent and can be adopted to confirm, or refute, HS linearity. In this study, \(\beta\) values and decay functions (all R\({}^{2}\geq\) 0.995) were estimated over four successive extractions on sample headspace from 1.500, 1.000, 0.500, and 0.100 g of EV olive oil. Results are reported in Table 3 for the 0.100 g sampling. It is worth noting that HS saturation occurred in the range between 0.500-1.500 g with relative \(\beta\) values of \(\sim\)1 for most of the analytes examined. Figure 6: Normalized response ratios (normalized volumes and % volumes) calculated between 0.1 and 1.5 g for selected analytes. Based on these data, any conjecture that is made as to the amount of analytes in a sample derived from HS-SPME sampling without having applied suitable quantitative strategies (e.g., MHE or SA) would be inconsistent and may lead to erroneous results. In practice, if internal standardization is used to estimate analyte concentrations in the sample using HS-SPME-TRIF sampling on 1.500 g, as reported in some research papers that deal with EV olive oil volatilome, relative errors would range between \(-90\%\) (for 1-octanol) to 1630% (for cyclohexane). These errors are just an underestimation of the actual ones, since additional sources of error, such as detection or response factors and chromatographic extra-column effects, are not computed. Only external calibration [42] or suitable flame ionization detector (FID) response-factor estimation [42; 43], accompanied by MHS-SPME or SA, can lead to the accurate quantitation of multiple analytes in samples headspace. Alternatively, sorption-based materials (PDMS or polyethylene glycol--PEG) with relatively high amounts of extraction polymer were able to overcome most of the limitations highlighted in the examples discussed herein. New commercial devices that benefit from the advantages of SPME, in terms of automation and instrument integration, such as SPME arrows [56; 57] and HisorbTM solutions (Markes International) deserve consideration.