## 4 Discussion
We can calculate the sum of slopes using the AS definition from Equation (2), and we get:  
\[\text{Sum of slopes}=\sum_{i=1}^{n-1}\frac{C_{i+1}-C_{i}}{t_{i+1}-t_{i}}=\sum_ {i=1}^{n-1}\frac{C_{i+1}-C_{i}}{\Delta t_{i}} \tag{4}\]  
where \(\Delta t_{i}\) refers to the sampling interval at time i. In the special case where the sampling interval is constant (\(\Delta t_{i}=\Delta t\)), at least for times up to Tmax, Equation (4) can be re-written:  
\[\text{Sum of slopes}=\sum_{i=1}^{n-1}\frac{C_{i+1}-C_{i}}{\Delta t} \tag{5}\]  
By eliminating the consecutive concentration values, we get:  
\[\text{Sum of slopes}=\frac{C_{n}}{\Delta t} \tag{6}\]  
where \(C_{n}\) is the last measured concentration up to the occurrence of the maximum, namely; that \(C_{n}\) refers to Cmax. In order to calculate AS from Equation (6), we have:  
\[\text{AS}=\frac{\text{Sum of slopes}}{\text{number of intervals}}=\frac{C_{n}}{n-1}=\frac{C_{max}}{\Delta t\,\times(n-1)} \tag{7}\]  
The product \(\Delta t\cdot(n-1)\) refers to Tmax, thus:  
\[\text{AS}=\frac{C_{\text{max}}}{T_{\text{max}}} \tag{8}\]Therefore, in the special case of equally spaced sampling intervals, AS becomes identical to Cmax/Tmax. When the sampling intervals are not equal, which is more realistic in actual BE studies, then AS (Equation (2)) offers the advantage of providing an estimation of slope based on many points (Figure 1b). In other words, AS is less dependent on the specific time point of Tmax, and therefore it is less influenced by its failure to accurately detect it due to sparse sampling. In the previous study [23], the Cmax/Tmax ratio was found to be a metric expressing the rate of absorption. Its properties have been assessed and compared to other PK parameters using machine learning. In that study, it was found that Tmax was the major contributor to Cmax/Tmax irrespective, of absorption kinetics. The current analysis extends previous findings by introducing the new metric, average slope, which is a generalization of Cmax/Tmax, and performing more simulations and machine learning analyses. The new metric, AS, succeeds in expressing the kinetic properties of absorption both conceptually (i.e., with the appropriate units and physical meaning) and in practice, namely, from the machine learning results. It should be underlined that AS can be easily calculated by applying a model-independent method using the average of the subsequent slopes (see Equation (2)). Additionally, it does not require any assumptions (e.g., first-order kinetics), any model (e.g., one- or two-compartment), or transformations. The estimation method of AS is as simple as that of the widely used AUCt in the case of the extent of absorption. In pharmacokinetics, the typical C-t profile (Figure 1) observed after oral administration is the result of three processes occurring concomitantly in the body: absorption, distribution, and elimination. Absorption dominates in the early time points, namely during the rapid increase in concentration values, while there is an equilibrium between the three processes as time passes. The time point where there is an equilibrium, namely the input rate (i.e., absorption rate), becomes equal to the output (i.e., elimination and/or distribution to tissues) refers to Tmax. Thus, in order to place more emphasis on absorption kinetics, a modified version of AS can also be defined.