# In Silico Modeling Study of Curcumin Diffusion and Cellular Growth
## 2 Methodology
### Fibroblast Culture Growth Model
\[\frac{\mathrm{d}\mathrm{M}\mathrm{d}}{\mathrm{d}t}=\mathrm{v}_{\mathrm{s} \mathrm{d}}\cdot\left(\frac{\mathrm{G}\mathrm{F}}{  
Next, to visualize the cell division process, we added a cellular Potts model to the simulation, consisting of a grid-based spatial representation that allows for cell description [41]. Additionally, cell division conditions were set based on a cycB threshold value: the G2-M transition initiates when cycB > 2.6 uM, and mitosis occurs when cycB > 2.9 uM. [PERSON] and [PERSON] reported that cycB allows the G2-M transition and controls the start of mitosis [32]; thus, a high enough concentration of cycB was selected to enable cell division to occur within a reasonable time frame. A probabilistic condition was defined to avoid a uniform division. This assumption helped create a more realistic model.  
The experimental conditions for the proliferation model, such as the number of initial cells and total simulation time, were defined based on a standard fibroblast culture and scratch-wound assay, as demonstrated in [12,14]. In this type of experiment, the wound-healing effect is assessed by scratching a line through the culture [2]. As the authors in [14] reported, wound healing occurs in two phases: cell migration and cell proliferation [14]. Cell migration occurred in the first 24 h of their experiment, whereas proliferation continued until the end of the trial at 72 h. As such, the proliferation model represents an experiment  
Figure 1: Simplified cell cycle model by [PERSON] and [PERSON] [32] representing the pathway of cyclin-dependent kinases (CDKs).  
Figure 2: Cell cycle protein oscillatory behavior, after running the model of [PERSON] and [PERSON] [28].  
of 72 h with around 50,000 initial cells [14]. However, due to the extensive computation time that it required, both the number of seeded cells and total simulation time had to be scaled down. Thus, 447 initial cells were seeded in a simulated 6.4 mm-diameter well from a Corning(r) 96-well microplate (Corning cat. no. CLS3595, Sigma, St. Louis, MO, USA). Similarly, the experimental time was halved and set to 36 h. Cell migration occurred in the first 12 h, then the cells began to proliferate until the end of the simulation for a total of 24 h.  
In order to properly visualize the wound-healing process, the well diameter was multiplied by a scaling factor of 100, generating a cell domain of 640 arbitrary units (a.u.). Fibroblast diameter has been described in the ranges 10-15 uM [42] and 15-25 uM [43]. In this work, we considered the seeded cells' diameter to be between 15 and 20 uM, with a doubling time between 24 and 32 h. The overall cell diameter range was shortened to limit confluence variations between simulations for the same case, whether control or curcumin dose, generated by cell area randomness. To calculate an area range for the seeded cells, the minimum and maximum diameter values were converted to millimeters and multiplied by a scaling factor of 100. Then, this interval was scaled by a factor of 100 to compensate for the reduction in seeded cells. The area was randomized for every cell before and after cell division, between 177 and 314 a.u. For simplicity, the cells were defined as circles.  
Finally, the scratch line was recreated by organizing the cells into six cell types: exterior, medial, and internal, for both left and right sides of the wound. Each cell population was defined by a rectangle of variable dimensions dependent on the scratch-line size, set to 132 a.u., which is six times the scratch wound in [14], scaled by a factor of 100. It is important to note that these changes represent only a visual scale and do not affect the overall wound-healing process. Additionally, external and medial cell populations were set to 150 initial cells, whereas internal cell populations were set to 60 initial cells. However, as cells are limited to existing within the well's domain, the initial number of cells was restricted to the following: on the left side, external (63), medial (106), and internal (57); on the right side, external (61), medial (107), and internal (53). The scratch-wound assay initial configuration is shown in Figure 3.  
Figure 3: Initial scratch-wound assay configuration at \(t=0\) h. Each rectangle contains a cell population that has a different cell migration and proliferation effect during wound healing. At this time, cells have not grown to a random cell area.  
As shown in [14], cell types move toward the wound during cell migration, depending on their proximity to the scratch line. Internal cells move faster than those closer to the scratch line, whereas external cells barely move from their initial position. For simplicity and to lower computational demands, only internal cells were set to follow the cell cycle and proliferate because it is expected for this cell type to cover the simulated wound after cell migration ends.  
In general, wound healing is measured in vivo as the extent of wound closure measured and compared to the original wound size [10; 44], or in vitro as the scratched area not covered by cells [14]. In the present work, curcumin's effect on wound healing was measured by calculating the confluence or the surface percentage of the well covered by adherent mammalian cells at any given time [45] because this was simpler to obtain and compute. Even though this was a general measurement of the well and not specific to the area of interest, it served the same purpose, as it is highly dependent on the number of cells entering mitosis due to curcumin's effect. Confluence was calculated using the sum of every cell area and the well area (321,700 a.u.), as follows:  
\[\text{C}=\frac{\text{Total cell area}}{\text{Well area}}\cdot 100\%. \tag{7}\]  
As previously stated, the cell area is randomly assigned to every cell (177-314 a.u.).