# In Silico Modeling Study of Curcumin Diffusion and Cellular Growth
## 3 Results and Discussion
### Curcumin Diffusion Model  
The percentages shown in Table 1 were used to calculate the concentration of curcumin released in time \(t\) for the six doses selected. Then, a K-P fit was applied with a correlation coefficient (\(R^{2}\)) of 0.8991. An average diffusion coefficient \(n\) value of 0.18 and \(K_{m}\) of 0.58, \(C_{1}\) = 0.9857, and \(C_{2}\) = 0.9457 were obtained. These values were similar for all concentrations since they dictate the release mechanism, which is dose-independent. A value of \(n<5\) signifies a diffusion release mechanism [61]. Figure 8 shows relative curcumin release for the lower concentrations (0.25, 0.5, and 1 uM), where it was normalized to the maximum release value of \(M_{\infty}\) = 0.91 ug. The K-P model (dashed line) underestimates earlier release values while overestimating later release values. Since it is not totally asymptotic, the fit surpasses the final curcumin release, which did not affect the final simulations. [PERSON]'s hyperbolic model (solid line) gives a slightly more accurate fit with some underestimation of the earlier releases. The results were compared to previous curcumin-release models [17; 53; 62], and a similar curve was observed. Nonetheless, the time for the total concentration of released curcumin varies. Several factors affect the kinetics of drug release, such as the administration mechanism (oral, topical) and physical properties inherent in the substance (solubility, molecular weight, partition coefficient) [18], as well as the composition of the matrix en  
Figure 7: Cell population growth for control group. At 27.1 h, cell divisions begin, gradually increasing until the cell population reaches 557 at 27.85 h. The cell population shows exponential growth. capsulating the drug [17]. These factors may affect the curcumin-release rate, and the cell division probabilities would vary. The diffusion-controlled release depends on the porosity of the polymeric matrix and the mechanical mass transfer through this matrix [17]. Therefore, although the release times are different, curcumin's release mechanism was appropriately modeled. The curves generated in Figure 8 have a similar form to those found in other studies [17; 53; 62]. Several studies have reported a curcumin-release ratio or release percentage of 90-100%, showing biphasic behavior [63; 64; 65; 66]. In the first phase, an initial burst occurs associated with the fast release of molecules located in the core interface of the polymer, whereas in the second phase, curcumin progressively releases from the inner core to the outer aqueous environment [63]. Rapid release has been reported to last up to 12 h, whereas curcumin release stabilizes at around 30 h [63; 64; 65; 66]. However, release rates depend on several factors, such as the polymeric matrix composition, diameter, and the pH of the aqueous environment [63; 64; 65; 66]. Many strategies for drug-delivery systems have been formulated to avoid the dumping phenomenon and provide desirable biphasic drug release mediated by diffusion. Initial burst release delivers the drug for distribution to a large volume to rapidly reach the therapeutic concentration, and a slow controlled release maintains the therapeutic concentration for prolonged periods [67]. Several in vitro studies have shown the hormetic behavior of curcumin in human skin cells. The effect of curcumin can be stimulatory or inhibitory, depending on the doses. Research on wound healing displayed that lower concentrations of curcumin stimulate the proteasome, increasing the differentiation of keratinocytes and inhibiting cell migration  
Figure 8: Curcumin release relative to final release for three different concentrations (5, 10, and 15% w/w). The error bars represent the standard deviation. The dashed line is the average Korsmeyer-Peppas model fitted to the data. The solid line is the average Peleg hyperbolic fitted model. The experimental values, means, and standard deviations obtained by [PERSON] et al. were used [17]. at higher doses.