## 3 Results and Discussion
### Analysis and Discussion of Release Results
The comparison between experimental data and model curves is shown in Figure 17. The small nonsystematic deviation between the model and data may be due to experimental uncertainties and it clearly suggests that it is worthless to seek for a more elaborate model. The coefficient KD found from fitting of the experimental curves appears to depend on the drug content (unlike in the case without clay). The values found are K\({}_{\text{D}}\) = 1.83 \(\times\) 10\({}^{-3}\) s\({}^{-1}\), 2.17 \(\times\) 10\({}^{-3}\) s\({}^{-1}\), and 2.5 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) for drug contents of 5 wt %, 10 wt %, and 20 wt % respectively. So, it appears that the ratio of clay to drug content has an effect on the diffusion kinetics of drug in the gel. Figure 16: Comparison between experimental (symbols) and model (continuous lines) release curves for several PEO and aprepitant formulations. Figure 17: Comparison between experimental (symbols) and model (continuous lines) release curves for several PEO/aprepitant/clay formulations. Clay weight fraction = 5 wt %. The release curves for the case of 5 wt % clay content in PEO matrix cannot be described by the first-order kinetic model. The deviation between the data and model is systematic, suggesting that there is a second release characteristic time which is much smaller than the primary one, which corresponds to molecular diffusion through the gel. An amount of drug is released by diffusion and another amount by combination of diffusion and dissolution in bulk liquid steps. So, a physically reasonable model which can describe the data must be developed. This physical model is the following: A fraction (1 - f) of the drug amenable to dissolution is in molecular form and a fraction f is in solid form (i.e., nanometer size particles). Both types of drug diffuse through the gel layer and reach the bulk liquid. Then, in the bulk liquid, the second fraction (solid particles) undergoes a slow dissolution process (similar to that of solid drug shown in Figure 16). The mathematical model describing the above scenario is:  
\[\frac{\mathrm{d}C_{1}}{\mathrm{dt}}=-K_{\mathrm{D}}((1-f)C_{\mathrm{F}}-C_{ 1}) \tag{3}\]  
\[\frac{\mathrm{d}X_{1}}{\mathrm{dt}}=-K_{\mathrm{D}}X_{1} \tag{4}\]  
\[\frac{\mathrm{d}X_{2}}{\mathrm{dt}}=K_{\mathrm{D}}X_{1}-k_{\mathrm{s}}X_{2} \tag{5}\]  
where \(X_{1}\) is the weight fraction of solid (amenable to dissolution) drug in the matrix and \(X_{2}\) is the fraction of solid drug located in the bulk liquid. The rate coefficient KDS corresponds to the diffusion of the solid particles through the gel layer and the rate coefficient k to particle dissolution. The bulk drug concentration can be found as:  
\(C=C_{1}+fC_{\mathrm{F}}(1-X_{1}-X_{2})\). The systems of the equation with initial conditions \(C_{1}=0\), \(X_{1}=1\), \(X_{2}=0\) is solved analytically to give:  
\[C=(1-f)C_{\mathrm{F}}(1-\exp(-K_{\mathrm{D}}t))+fC_{\mathrm{F}}(1-\exp(-K_{ \mathrm{D}}s)-\frac{K_{\mathrm{D}}}{k_{\mathrm{s}}-K_{\mathrm{D}}}(\exp(-K_{ \mathrm{D}}s)-\exp(-k_{\mathrm{s}}t))) \tag{6}\]  
The above equation must be fitted to the experimental data by appropriate choice of \(K_{\mathrm{D}}\), \(K_{\mathrm{D}}\), \(k_{\mathrm{s}}\), and f. A very good fitting was achieved by considering a single set of parameters (assuming drug content-independent parameter values). The comparison between the data and model curves is shown in \[\rm C=C_{eq}(1-exp(\frac{Ak}{V}t)) \tag{2}\]0 The fitting parameters are f = 0.11 (i.e., 11% of the finally dissolved drug is initially in solid form), \(K_{\mathrm{D}}=2.5\times 10^{-3}\) s\({}^{-1}\) (diffusion through gel layers is somewhat faster than in the absence of clay), \(K_{\mathrm{D}}=0.82\times 10^{-3}\) s\({}^{-1}\) (i.e., diffusion of particles is much slower than the diffusion of molecular drug), and \(k_{\mathrm{s}}=0.8\times 10^{-3}\) s\({}^{-1}\). The rate constants found are compatible with the physics of the problem.