# Average Slope vs. Cmax: Which Truly Reflects the Drug-Absorption Rate?
## 3 Results
### Comparative Performance between AS and Cmax
#### 3.2.1 Principal Components Analysis  
In order to find insights from each dataset and explore the interconnections between the PK parameters, PCA was utilized. The orange dots in the plane generated from the two primary PC observations represent simulated data for each of the 1000 individuals. In the meantime, the lines depict the vectors of the variables, including AS, AUC, AUCinf, Cmax, and Tmax. On the right side of each figure, the loadings of all PK parameters are displayed, where PC1 and PC2 refer to the loadings of the first and second PC, respectively. Complementary to the PCA, scree plots were constructed to identify the ideal number of PCs. The eigenvalues (representing the proportion of variance explained) are depicted on the y-axis, and the number of components is displayed on the x-axis.  
Figure 1 presents the results of the PCA analysis in the case of lisdexamfetamine. It can be observed in Figure 1, specifically in the left plot, that _AUC_ and _AUCinf_ are closely positioned on the upper right part of the panel, adjacent to the first principal component, with similar l1 values (0.34 for _AUCinf_ and 0.39 for _AUC_). Additionally, Cmax (l1 = 0.58) is positioned beside AS (l1 = 0.55), indicating a strong association between them. Tmax, currently regarded as the parameter best reflecting absorption kinetics, is situated at the upper left side of the panel. Notably, the vector representing AS is diametrically opposite  
to Tmax, while the two PK parameters form an angle of a little less than 180\({}^{\circ}\) degrees, suggesting a contrasting kinetic behavior between them.  
Figure 2 illustrates the PCA results for the remaining four drugs, namely nintedanib, methylphenidate, nitrofurantoin, and theophylline. A visual inspection of Figure 2 shows that _AUC_ and _AUCinf_ are positioned closely together in every PCA analysis, regardless of the absorption kinetics of each drug. The Cmax in every analysis lies between AS and the pair of AUC and AUCinf, indicating a similar kinetic behavior with these PK variables. Significantly, the vector representing AS is diametrically opposite to Tmax, with the angle between them being above 90\({}^{\circ}\) for every PK profile. This indicates that as AS increases, reflecting faster absorption, Tmax decreases. Therefore, AS can effectively capture changes in absorption kinetics. This cannot be implied for Cmax because it forms an angle with Tmax, which is between 90\({}^{\circ}\). The corresponding scree plots are depicted in Figure A1.  
Figure 1: Principal component analysis of the pharmacokinetic parameters of lisdexamfetamine. Left illustration: biplot of the two principal components with the individual scores (orange dots) and the loadings (blue lines) of the pharmacokinetic parameters. Central illustration: scree plot to identify the number of principal components. Right illustration: loading values for the two initial principal components.  
Figure 2: Four different principal component analyses were conducted, each referring to one of the additional four drugs in the study. Each plot corresponds to (**a**) nintedanib, (**b**) methylphenidate,  
(**c**) nitrofurantoin, and (**d**) theophylline. Left panel of each set: biplot of the two principal components showing the individual scores (orange dots) and the loadings (blue lines) of the pharmacokinetic parameters. Right panel of each set: loading values for the two initial principal components.  
The high explained variance values indicate the strong descriptive capability of all the PCA models developed. Specifically, for isdexamfetamine (Figure 1), the first and second principal components accounted for 91.97% of the total variability (49.33% and 42.64%, respectively). The total explained variability was 91.35% for nintedanib, 97.22% for methylphenidate, 98.91% for nitrofurantoin, and 98.80% for theophylline (Figure 2).