# In Silico Modeling Study of Curcumin Diffusion and Cellular Growth
## 2 Methodology
### Integration of Fibroblast Proliferation and Curcumin Diffusion Models  
In a comparative evaluation study on the effects of curcumin and chlorheoxidine on human fibroblast viability and migration, the lowest concentration of curcumin (0.003%) exhibited low fibroblast cytotoxicity and excellent wound-healing properties [12]. A scratch-wound assay was performed, and the percentage of fibroblast migration and proliferation was measured and compared to a control group in different time frames: 24, 48, and 72 h [12]. It was observed that for every time step, the population had an average growth of 3.4 times its initial value. Since in this study we are using a cellular division model, we estimated the equivalent cellular division period by the equation:  
\[P_{0}+FP_{0}=(P_{0})\cdot 2^{\frac{t}{\tau}}, \tag{10}\]  
where \(P_{0}\) is the initial cell population, \(F\) is 3.4, \(t\) is the original period where cell division occurs, and \(\tau\) is the period for the expected target population in time \(t\). Thus, to obtain a new period for cell division to occur, considering 3.4 as the maximum effect, the culture was treated with the lowest concentration of curcumin selected (0.25 uM) [12; 15]. Equation (5) suggests that the cell population exhibits exponential growth by a factor of 2, and the expected population is the initial population plus the new population increased by a factor of \(F\), representing wound healing. When solving for \(\tau\), a value of 12.69 h was obtained; therefore, cell divisions were expected to occur every 12.69 h with a curcumin concentration of 0.25 uM. The model's parameters must be altered to modify the cell division period, since they control the rates of synthesis and degradation of the different complexes, regulating the cell cycle oscillations. A multiplier variable \(m\) was defined in this study, allowing the parameters to be modified for the curcumin concentration. Therefore, the parameters of the model were redefined in Morpheus as functions and multiplied by \(m\). \[m=-0.1139M+2.4135. \tag{11}\]  
The value of \(m\) is defined by Equation (6), which directly relates to the concentration of curcumin released (\(M\)) following the behavior defined by the K-P and Peleg equations (Equations (3) and (4)). In order to determine the former equation, four key points were defined: with a dose of 0.25 uM, the cell division period must be 12.69 h; with a dose of 10 uM, almost no effect is visible [15; 16; 46]; with curcumin at 15 uM, negative effects are visible, and the healing process is slowed [14; 47; 55]; with a 20 uM concentration, cytotoxic effects are visible, and no cell divisions occur [14; 46; 47; 48; 49]. With these considerations, \(m\) values were calibrated for each concentration until the desired effects were obtained, generating an approximately linear equation describing curcumin's dose-dependent effect, as shown in Figure 4.  
This healing process was also simulated by varying the final cell confluence, ultimately modifying the total cell numbers. [PERSON] and [PERSON] [14] demonstrated that a cell culture treated with the lower dose range (0.25, 0.5, 1, and 5 \(\upmu\)M) showed greater wound-healing effects, 0.25 \(\upmu\)M being the best therapeutic dose. Conversely, a dose of 20 \(\upmu\)M was inhibitory and even resulted in apoptotic cell death. This work altered the total cell population to recreate this dose-dependent stimulation and inhibition. This was achieved by modifying the probability of cell division \(p\), which was evaluated whenever cycB reached the 2.9 \(\upmu\)M threshold. A number between 0 and 1 was randomly generated; if it was less than the probability, the cell would divide. Three probabilities were assigned depending on curcumin concentration: \(p_{1}=5\times 10^{-4}\), \(p_{2}=5\times 10^{-5}\), and \(p_{3}=5\times 10^{-6}\). The probabilities used for each curcumin concentration are summarized in Table 2. The combination of probabilities depends on the curcumin concentration and the desired effect on cell proliferation.