## 3 Results and Discussion
### Models for Prediction of FPF  
The FPF prediction model performance is summarized in Table 2. Both _fugeR_ and _rgp_ models exhibited good FPF predictive ability. Fuzzy systems resulted in a slightly worse performance in comparison to genetic programming, as their NRMSE was 8.86 with \(R^{2}\) below 0.85. The best model was developed using _rgp_ package and it is presented as Equation (4) (NRMSE = 8.29%, and \(R^{2}\) = 0.91).  
Figure 2: The workflow of in-house developed software implementing _rgp_ framework to model development and assessment system using genetic programming methods. The first stage is based on a symbolic regression function allowing the development of formal mathematical equation fitting defined data. Later, the process is stopped, models are tested according to the k-fold cross-validation (k-CV) scheme, and the evolution process is resumed. Parameters affecting the evolutionary algorithms and model development process can be modified in the text user interface [19].  
The model uses four variables: API to excipient ratio (X\({}_{1}\)), feedstock solution concentration (X\({}_{2}\)), ethanol to water ratio (X\({}_{3}\)), inlet air temperature (X\({}_{4}\)), and five constants (C\({}_{1}\)-C\({}_{5}\)). During the evolution of the solution, two variables describing flow rate and pressure inside the spray drying chamber were excluded. Comparison of FPF values predicted by the model and those obtained in laboratory experiments is presented in Figure 3.  
\[\text{FPF}(\%)=\sqrt{\left[\text{C}_{1}\cdot\text{X}_{4}+\text{X}_{2}^{2} \cdot\sqrt{\left(\text{X}_{3}\right)}+2\cdot\text{X}_{1}+\text{C}_{2}\right] \cdot\left[\sqrt{\left(2\cdot\text{X}_{4}+\text{X}_{2}^{2}+\text{C}_{3}\right)} +\text{C}_{4}\right]+\text{C}_{5}} \tag{4}\]  
where:  
C\({}_{1}\)-C\({}_{5}\): constants; C\({}_{1}\) = 5.298; C\({}_{2}\) = -524.223; C\({}_{3}\) = -135.864; C\({}_{4}\) = -13.842; C\({}_{5}\) = 120.661,  
X\({}_{1}\): API to excipient ratio [m/m%],  
X\({}_{2}\): concentration of feed solution [m/V%],  
X\({}_{3}\): ethanol to water ratio in solvent applied in the process [V/V%],  
X\({}_{4}\): inlet air temperature during spray drying process [\({}^{\circ}\)C],  
M: mass,  
V: volume.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **RMSE** & **\(\text{R}^{2}\)** & **NRMSE [\%]** \\ \hline Im() & 12.03 & 0.62 & 20.44 \\ _fugeR_ & 5.21 & 0.84 & 8.86 \\ _rgp_ & 4.88 & 0.91 & 8.29 \\ \hline \hline \end{tabular} FPF—fine particle fraction; RMSE—root-mean-square error; \(R^{2}\)—coefficient of determination; NRMSE—normalized root-mean-square error.  
\end{table}
Table 2: Summary of FPF prediction performance for models created with different methods.  
Figure 3: Comparison of FPF values predicted by the model and obtained in laboratory experiments. FPF—fine particle fraction.  
Analysis of Figure 5A shows that changing the ratio of API to excipient content strongly impacts ED and FPF values. Increasing the API content in the liquid feed decreases both the FPF and ED. The steepest descent of ED is observed when API content is in the range of 50 to 72.5% (Figure 5A). Figure 5B presents the influence of the total concentration of compounds dissolved in feed solution on the final particle characteristics. It is observed that lower concentrations of feed solution yield higher FPF and ED values. Ethanol to water ratio in solvent applied in the process has a small impact on FPF value while predicted ED is significantly lower for feed solvents with lower ethanol content (Figure 5C). The temperature of the air entering spray drying chamber does not affect ED at all (Figure 5D). On the contrary, low and high temperatures positively impact FPF values (Figure 5D).