In Silico Modeling Study of Curcumin Diffusion and Cellular Growth

Abstract

Curcumin can enhance cutaneous wound healing by improving fibroblast proliferation. However, its therapeutic properties are dose-dependent: high concentrations produce cytotoxic effects, whereas low concentrations benefit cell proliferation. Similarly, the type of administration and its moderation are key aspects, as an erroneous distribution may result in null or noxious activity to the organism. In silico models for curcumin diffusion work as predictive tools for evaluating curcumin’s cytotoxic effects and establishing therapeutic windows. A 2D fibroblast culture growth model was created based on a model developed by Gérard and Goldbeter. Similarly, a curcumin diffusion model was developed by adjusting experimental release values obtained from Aguilar-Rabiela et al. and fitted to Korsmeyer–Peppas and Peleg’s hyperbolic models. The release of six key curcumin concentrations was achieved. Both models were integrated using Morpheus software, and a scratch-wound assay simulated curcumin’s dose-dependent effects on wound healing. The most beneficial effect was achieved at 0.25 μM, which exhibited the lowest cell-division period, the highest confluence (~60% for both release models, 447 initial cells), and the highest final cell population. The least beneficial effect was found at 20 μM, which inhibited cell division and achieved the lowest confluence (~34.30% for both release models, 447 initial cells). Confluence was shown to decrease as curcumin concentration increased, since higher concentrations of curcumin have inhibitory and cytotoxic effects.

1. Introduction

Curcumin is the main bioactive polyphenolic component derived from Curcuma longa, a plant typically harvested in Southwest Asia and widely used for medical purposes. Multiple studies have shown that curcumin has anti-inflammatory, antiseptic, and antioxidant effects and exhibits anticancer activity [1]. Curcumin’s healing potential is attributed to its biological activities, which enhance cutaneous wound healing by improving epithelial regeneration, fibroblast proliferation, and vascular density [2,3]. Due to its therapeutic properties, this molecule is being researched as a potential treatment for burns. Oxidative stress is a factor that negatively impacts the cicatrization process of injuries such as burns [4]. This phenomenon occurs when an imbalance exists between the production and clearance of reactive oxygen species (ROS), eventually leading to tissue damage [5,6,7]. Antioxidant therapies are of special interest in treating burns, as they could neutralize and reduce oxidative damage [6,7,8]. Curcumin’s antioxidant properties can be attributed to its chemical structure. Because it contains a variety of functional groups, curcumin can interact with diverse molecular mechanisms to reduce oxidative stress levels [7,8,9]. As an antioxidant agent, curcumin accelerates the wound-healing process by decreasing the damage caused by oxygen free radicals [10].

Thus, this molecule has been described as a possible substitute for existing commercial treatments due to its accessibility, lower cost, and lower cytotoxicity [11,12]. However, its therapeutic properties have been demonstrated to be dose-dependent [12,13,14,15]. When administered in high concentrations, it has cytotoxic effects on the cell population, reducing its viability and growth [12,13,14,15,16]. Moreover, the administration type and its moderation are essential since the drug delivery depends on the molecule’s properties [17,18]. Curcumin exhibits low solubility, low bioavailability, and fast metabolism [19]. An erroneous distribution may lead to secondary negative effects, as the molecule reaches sites outside its therapeutic window, causing null or noxious activity to the organism [18].

Even though there are reports of curcuminoids being classified as PAINS, in silico investigations have validated their binding and activity on specific proteins (casein kinase 1 and glycogen synthase kinase 3 beta) involved in the Wnt/β-catenin signaling pathway [20]. Molecular docking calculates curcumin derivatives’ binding affinity to epidermal growth factor receptor and nuclear factor κB (NF-κB), known as cancer target proteins [21]. In most cases, the in silico results have been exemplarily validated using in vitro assays accordingly.

Computational modeling is a widely used tool in biological research for prediction purposes and to guide experimental research [22]. Variables are used in these models to describe a simplified biological system or physiological process and are modified in simulations to visualize predictions for multiple hypothetical scenarios [23]. Wound healing and cell proliferation are extensively modeled processes [23], especially in respect of their behavior or response to the presence of a key variable. For instance, the authors of [24] proposed a model of the effects of epidermal growth factor, a protein secreted by fibroblasts, on keratinocyte migration. They used the cellular Potts model (CPM) to predict the mechanisms of keratinocyte migration and its effects on reepithelialization. The authors of [25] developed a model simulating the fibroblast network after myocardial infarction that predicts fibroblast signaling and wound healing.

Similarly, in silico models in drug development have demonstrated their ability to generate reliable predictions for drugs’ modes of action and the mechanisms underlying their side effects [26]. This approach has proven useful in optimizing drug dosage. By integrating in vitro cell-toxicity data with multiscale in silico modeling of drug exposure, these models may become efficient tools for assessing and predicting drug toxicity [26,27]. This approach allows a better understanding of biological responses and reduces uncertainties, allowing for better prediction of effective treatment [26]. Since curcumin’s therapeutic effects are dose-dependent, in silico models can help establish therapeutic windows and predict wound-healing effects at different doses.

Therefore, we aimed to develop an in silico model that integrates curcumin diffusion and fibroblast culture growth to identify curcumin’s cellular proliferation and wound-healing effects. This 2D model may serve as a predictive tool for calculating the growth of an in vitro fibroblast culture proportional to the delivered curcumin concentration. As previously described, curcumin’s cytotoxic effect must be evaluated to establish its therapeutic window. The results obtained provide a foundation for in vitro experimentation using only the curcumin dosage that causes the desired effect.

2. Methodology

2.1. Fibroblast Culture Growth Model

The final 2D model integrated two stand-alone models developed in the specialized software Morpheus version 2.2 to study multicellular systems [28]. Firstly, the fibroblast culture growth model was created. The variables were defined according to previous experimental and modeling studies [29,30,31,32,33,34,35,36,37,38,39]. Two approaches were identified based on cell division: (1) controlled by probability and (2) controlled by the cell cycle. The second approach was selected for this work, as it considers the biological processes that rule cell mitosis. The model for the network of cyclin-dependent kinases driving the mammalian cell cycle, developed by Gérard and Goldbeter [32], was selected as the foundation for the proliferation model. The.sbml archive for the model was obtained from the BioModels repository due to Maire, Zyoud, and Nguyen [40].

Gérard and Goldbeter’s model consists of 5 variables and 24 parameters linked by a set of 6 differential equations (Equations (1)–(6)) that generate the cyclin-dependent kinases’ oscillatory behavior.

$$\frac{\mathrm{dMd}}{\mathrm{dt}}={\mathrm{v}}_{\mathrm{sd}}·\left(\frac{\mathrm{GF}}{{\mathrm{K}}_{\mathrm{gf}}+\mathrm{GF}}\right)-{\mathrm{V}}_{\mathrm{dd}}·\left(\frac{\mathrm{Md}}{{\mathrm{K}}_{\mathrm{dd}}+\mathrm{Md}}\right),$$

(1)

$$\frac{\mathrm{dE}2\mathrm{F}}{\mathrm{dt}}={\mathrm{v}}_{1\mathrm{e}2\mathrm{f}}·\left(\frac{\left(\mathrm{E}2{\mathrm{F}}_{\mathrm{tot}}-\mathrm{E}2\mathrm{F}\right)}{{\mathrm{K}}_{1\mathrm{e}2\mathrm{f}}+\left(\mathrm{E}2{\mathrm{F}}_{\mathrm{tot}}-\mathrm{E}2\mathrm{F}\right)}\right)·\left(\mathrm{Md}+\mathrm{Me}\right)-{\mathrm{V}}_{2\mathrm{e}2\mathrm{f}}·\left(\frac{\mathrm{E}2\mathrm{F}}{{\mathrm{K}}_{2\mathrm{e}2\mathrm{f}}+\mathrm{E}2\mathrm{F}}\right)·\mathrm{Ma},$$

(2)

$$\frac{\mathrm{dMe}}{\mathrm{dt}}={\mathrm{v}}_{\mathrm{se}}·\mathrm{E}2\mathrm{F}-{\mathrm{V}}_{\mathrm{de}·}\mathrm{Ma}·\left(\frac{\mathrm{Me}}{{\mathrm{K}}_{\mathrm{de}}+\mathrm{Me}}\right),$$

(3)

$$\frac{\mathrm{dMa}}{\mathrm{dt}}={\mathrm{v}}_{\mathrm{sa}}·\mathrm{E}2\mathrm{F}-{\mathrm{V}}_{\mathrm{de}}·\mathrm{Cdc}20·\left(\frac{\mathrm{Ma}}{{\mathrm{K}}_{\mathrm{da}}+\mathrm{Me}}\right),$$

(4)

$$\frac{\mathrm{dMb}}{\mathrm{dt}}={\mathrm{v}}_{\mathrm{sb}}·\mathrm{Ma}-{\mathrm{V}}_{\mathrm{db}}·\mathrm{Cdc}20·\left(\frac{\mathrm{Mb}}{{\mathrm{K}}_{\mathrm{db}}+\mathrm{Mb}}\right),$$

(5)

$$\frac{\mathrm{dCdc}20}{\mathrm{dt}}={\mathrm{v}}_{1\mathrm{cdc}20}·\mathrm{Mb}·\left(\frac{\left(\mathrm{Cdc}{20}_{\mathrm{tot}}-\mathrm{Cdc}20\right)}{{\mathrm{K}}_{1\mathrm{cdc}20}+\left(\mathrm{Cdc}{20}_{\mathrm{tot}}-\mathrm{Cdc}20\right)}\right)-{\mathrm{V}}_{2\mathrm{cdc}20}·\left(\frac{\mathrm{Cdc}20}{{\mathrm{K}}_{2\mathrm{cdc}20}+\mathrm{Cdc}20}\right).$$

(6)

The model involves four cyclin/cyclin-dependent kinases (cyclin/CDKs) complex variables that progress through the cell cycle’s successive phases. Other variables include the transcription factor E2F and the protein Cdc20. Since the cyclin D/CDK4-6 level is only regulated by the growth factor (GF) level, it rapidly reaches a steady state. Therefore, the model contains only five variables [32]. The variables’ interaction with the model is shown in Figure 1 and can be summarized as follows:

• Cyclin D–CDK4-6 complex (cycD): GF directly activates the synthesis of cycD. The cyclin D–CDK4-6 complex initiates the cell cycle by promoting cell progression from the resting phase (G0) to the G1 phase.
• Transcription factor E2F (E2Fa) activates the synthesis of cyclin E–CDK2 and cyclin A–CDK2 complexes, and also actively participates in DNA synthesis.
• The cyclin E–CDK2 complex (cycE) strengthens the activation of E2F and allows G1–S transition.
• The cyclin A–CDK2 complex (cycA) degrades the cyclin E–CDK2 complex during the S phase. It also allows S–G2 transition by deactivating E2F and promoting cyclin B–CDK1 complex synthesis.
• The cyclin B–CDK1 complex (cycB) allows G2–M transition, controlling the start of mitosis (a cell divides into two daughter cells). It activates the Cdc20 protein via phosphorylation during mitosis.
• Cdc20 protein (cdc20a): The main actor in the model’s negative feedback promotes the degradation of cyclin A–CDK2 and cyclin B–CDK1 complexes. It allows the completion of the cell cycle, and another cell cycle occurs in the presence of GF.

As a starting point, we used in the simulation the base model of Figure 1, which was obtained from the BioModels repository due to Maire, Zyoud, and Nguyen [40]. Running this model demonstrates the oscillatory behavior of the cell cycle proteins, as shown in Figure 2.

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 μM, and mitosis occurs when cycB > 2.9 μM. Gérard and Goldbeter 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 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^® 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 μM [42] and 15–25 μM [43]. In this work, we considered the seeded cells’ diameter to be between 15 and 20 μM, 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.

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:

$$\mathrm{C}=\frac{\mathrm{Total}\mathrm{cell}\mathrm{area}}{\mathrm{Well}\mathrm{area}}·100\%.$$

(7)

As previously stated, the cell area is randomly assigned to every cell (177–314 a.u.). The confluence falls within a range dependent on the total cell area. Moreover, since curcumin has a dose-dependent effect on cell proliferation and the final cell population, the possible confluence range generated varies for every concentration. This range was obtained as the minimum and maximum confluence for two extreme cases, where every cell has an area of 177 or 314 a.u., respectively, as shown in

Table 1. Confluence range values for different curcumin concentrations.

	Concentration (μM)
Model	0.25	0.5	1	5	10	20
K-P 1	42.75–75.84%	37.52–66.57%	34.61–61.40%	32.46–57.59%	30.65–54.37%	24.59–43.63%
HB 2	42.70–75.74%	42.70–75.74%	34.11–60.52%	33.29–59.05%	30.65–54.37%	24.59–43.63%

¹ Korsmeyer–Peppas. ² Peleg’s hyperbolic.

.

2.2. Curcumin Diffusion Model

According to several studies, the threshold of curcumin’s therapeutic properties was defined at 10 μM as it showed minimal effect on cellular growth. Doses above this concentration proved to have inhibitory effects on cellular migration and proliferation [14,46,47,48,49], while doses below this concentration were more effective in stimulating cell proliferation [14,46,50]. Therefore, in this study, concentrations below 10 μM were considered low doses, while concentrations above it were classified as high doses. After considering the doses and their effects shown in [14] and this threshold, six different concentrations were selected to investigate both the positive and negative effects of curcumin on fibroblast proliferation. These concentrations were 0.25, 0.5, 1, 5, 10 and 20 μM.

To model curcumin diffusion in this study, the experimental values obtained by Aguilar-Rabiela et al. were used [17]. In that experiment, the authors modeled the release of curcumin from polyhydroxybutyrate (PHB) microparticles using three different curcumin concentrations: 5%, 10%, and 15%.

These percentages were used to calculate the concentration of curcumin released for each of the six concentrations selected with the same time step as the authors [17]. The obtained values were fitted to the Korsmeyer–Peppas (K-P) equation (Equation (3)) and Peleg’s hyperbolic (HB) model [51,52] (Equation (4)). Various studies have found a high correlation between the K-P model and the experimental results of curcumin release [53,54], as well as Peleg’s hyperbolic model.

$$\frac{M_t}{M_{\infty }}=K_m{t}^n,$$

(8)

$$\frac{M_t}{M_{\infty }}=\frac{C_{1}t}{1+C_{2}t}.$$

(9)

In both equations, $M_t$ is the amount of curcumin (µg) released at time t (hours), $M_{\infty }$ is the maximum amount of curcumin released (µg), $K_m$ is a rate constant for the exponential model, and n is the diffusional exponent of the model. For the hyperbolic model, $C_1$ is the release rate at the very beginning (1/h), and $C_2$ is the constant related to the maximum release (1/h). $C_1/C_2$ is also known as the capacity constant [52].

2.3. Integration of Fibroblast Proliferation and Curcumin Diffusion Models

In a comparative evaluation study on the effects of curcumin and chlorhexidine 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:

$$P_0+F{P}_0=\left(P_0\right)·2^{\frac{t}{\tau }},$$

(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 μM) [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 μM.

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.

$$m=-0.1139M+2.4135.$$

(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 μM, the cell division period must be 12.69 h; with a dose of 10 μM, almost no effect is visible [15,16,46]; with curcumin at 15 μM, negative effects are visible, and the healing process is slowed [14,47,55]; with a 20 μM 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. Demirovic and Rattan [14] demonstrated that a cell culture treated with the lower dose range (0.25, 0.5, 1, and 5 μM) showed greater wound-healing effects, 0.25 μM being the best therapeutic dose. Conversely, a dose of 20 μM 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 μM 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. Combination of probabilities defined for the left and right interior cell types for each curcumin concentration.

Concentration (μM)	Cell Type
	Left Interior	Right Interior
0 (control)	p1	p1
0.25	p1	p1
0.5	p1	p2
1	p2	p2
5	p3	p2
10	p1	p1
20	p3	p3

. The combination of probabilities depends on the curcumin concentration and the desired effect on cell proliferation. As curcumin concentration increases, cell division becomes less probable, simulating the dose-dependent effect of curcumin on cell proliferation. These probabilities apply only to the two cell types closest to the wound (left and right interior), which are ruled by Gérard and Goldbeter’s model for the cell cycle [32].

The cell division probabilities were fitted to reproduce observed wound-healing enhancements from curcumin in previous experiments [14]. Therefore, the model is specific to low doses of curcumin.

3. Results and Discussion

3.1. Fibroblast Culture Growth Model

The fibroblast proliferation simulation resulted in a doubling time of 27.1 h (Figure 5) found within the expected time range of 24–32 h. This initial doubling time can be altered depending on the experimental conditions and cell line depicted, as studies report varying times [56,57,58]. To obtain this period, the original model parameters were multiplied by a factor of ⅔. In addition, the cdc20 concentration was lowered to 4 μM to avoid a strong inhibitory effect in response to parameter manipulation.

Figure 6a shows the initial cell arrangement, randomly generated by the software. The simulated scratch wound is visible. Green and blue were used to represent the G2–M transition for cycB > 2.6 μM and cycB > 2.9 μM, respectively. Every other phase of the cell cycle (cycB < 2.6 μM) is also represented by green, since the generated images can only show the change in a single variable (in this case, cycB). Note that only the innermost cells are colored, since only the two cell types closest to the wound are affected by the cell cycle and will therefore proliferate. Figure 6b shows the cell arrangement after 12 h, where some of the innermost cells invaded the scratch wound and simulated cell migration to the wound site. The 447 seeded cells achieved an initial confluence of 34.28%.

After 12 h, the cell migration phase was complete, following the same behavior exhibited in [14], where cell migration occurred prior to cell proliferation in the first 24 h. The authors of [14] observed that cellular migration was dictated by the orientation of the cells, as those perpendicular to the wound were the first to migrate, followed by those parallel to it. This behavior was not recreated in the present model, since fibroblast shape was simplified to a circle. Instead, the proximity of the cell to the wound area is what determined the migration. Moreover, fibroblasts began to form a bridge in the wound [14]. Single-cell migration was modeled in the present model; however, the strength of the interactions between cells obliged them to cluster, forming structures similar to the cell bridges.

Entering the proliferation phase, when cycB > 2.9 μM, cell division conditions were met, and each cell’s probability to divide was evaluated according to the selected curcumin concentration and cell type (as shown in Table 2). After 27.85 h, every internal cell had divided, reaching a final population of 557 cells, as shown in Figure 7. The final confluence was 42.82%. Due to the probabilistic conditions, the internal cell population gradually entered mitosis, doubling from 110 to 220 cells. Figure 7 shows this behavior. This exponential growth occurs in diverse fibroblast cell lines, as demonstrated in various studies [58,59,60]. The results from this simulation were used as the control group.

3.2. 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, C₁ = 0.9857, and C₂ = 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 μM), where it was normalized to the maximum release value of $M_{\infty }=0.91\mathsf{\mu }\mathrm{g}$. 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. Peleg’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 encapsulating 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 at higher doses. A similar dose response of curcumin on the proteasome function has been demonstrated in other cell types, such as fibroblasts. Induction of the activity of the Na/K ATPase pumps in normal human epidermal keratinocytes treated with curcumin was evaluated. Lower doses stimulated the pump activity; meanwhile, a higher dose was inhibitory [68].

The present diffusion model shows the biphasic behavior developed using the experimental data in [17]. This behavior is of great importance in experimental settings, whether in vitro or in vivo, where curcumin is released in a controlled manner. Since curcumin has low solubility, low bioavailability, and fast metabolism [19], its release must be controlled to benefit the organism. In experimental settings, this can be achieved by encapsulating the active agent in a polymeric matrix, allowing for controlled dosage and release, thus extending the time that the therapeutic dosage is effectively present [18]. However, this model was fitted to a specific experiment, where the polymeric matrix comprises PHB microparticles. The release of curcumin can be better modeled if more experimental data are used with similar diffusion mechanisms.

3.3. Integration of Fibroblast Proliferation and Curcumin Diffusion Models

The six selected concentrations were simulated using the integrated model to visualize the effect of curcumin on wound healing.

Table 3. Time of cell division and final cell population for each curcumin concentration for 447 initial cells, where d1 and d2 are the times of the first and second divisions, respectively.

Korsmeyer–Peppas Model
Concentration (μM)	d1 (h)	d2 (h)	Final Cell Population (Number of Cells)
0 (control)	27.10	-	557
0.25	12.95	28.35	777
0.5	13.00	28.55	682
1	13.20	29.05	629
5	14.90	34.05	590
10	18.40	-	557
20	-	-	447 (Same as initial)
Peleg’s Hyperbolic Model
Concentration (μM)	d1 (h)	d2 (h)	Final Cell Population (Number of Cells)
0 (control)	27.10	-	557
0.25	12.95	28.35	776
0.5	13.05	28.60	715
1	13.20	29.05	620
5	15.00	34.10	605
10	18.75	-	557
20	-	-	447 (Same as initial)

summarizes the effect of curcumin on the cell division period and the final cell population obtained for each dose of curcumin and release model. All simulations had an initial cell population of 447. The lowest cell division period and highest final cell population were obtained at 0.25 μM. For both models, this period was similar to the previously calculated time frame (12.69 h), with a margin of error of 2%. Therefore, a dose-dependent effect of 3.4 was achieved.

The administration of low doses of curcumin (0.25, 0.5, 1, and 5 μM) allowed for two cell divisions to occur, resulting in a higher final cell population than the control. Notably, the second cell division was slower than the control’s first cell division for these concentrations. Since cell cycle oscillations are not perfectly defined as a sine or cosine wave, the time period for cell division is not exact. As shown in Figure 5, there is a difference between the time that CycB’s concentration is low and when it peaks. Thus, a time-delay effect exists between cell divisions.

As expected, for a concentration of 10 μM, the only beneficial effect is reducing cell division time. Conversely, the highest concentration of curcumin (20 μM) showed the lowest final cell population, as no cell division occurred. This same dose-dependent effect can be observed for both release models, with minimal variation in cell division time and final cell population. Hence, both curcumin-release models are viable to use for this application.

Figure 9 summarizes the effect of curcumin on confluence for both curcumin diffusion models. For the K-P model in Figure 9a, the best benefit, obtained at a concentration of 0.25 μM, represented the highest confluence (59.90%). Similarly, low concentrations (0.5, 1, and 5 μM) show curcumin’s beneficial effects, with a higher confluence than the control (52.39, 48.44, and 45.33%, respectively). For a 10 μM concentration, a similar confluence to the control was obtained (42.80%). Higher concentrations show inhibitory and cytotoxic effects: a dose of 20 μM negatively impacted cell division time; thus, the internal cells did not enter mitosis within the 36-hour time frame, resulting in the lowest confluence (34.28%).

For Peleg’s hyperbolic model in Figure 9b, the same dose-dependent effect was observed, and the confluence results showed only slight variations attributable to randomly defined fibroblast diameters in each simulation in the 15–20 μM range. The most beneficial effect was observed at a concentration of 0.25 μM with a confluence of 59.78%. Low concentrations (0.5, 1, and 5 μM) showed a higher confluence than the control (55.04, 47.55, and 46.19%, respectively), whereas a medium concentration (10 μM) showed no difference from the control (42.83%). The highest concentration (20 μM) resulted in no cell division and the lowest confluence (34.27%). Therefore, as described in Table 3, the same dose-dependent tendency is observed, independently of the curcumin diffusion model.

The dose-dependent effect of curcumin on confluence is summarized in Figure 10. Considering a 42.82% confluence as the control, the lowest concentration of curcumin demonstrated the most beneficial effect and represented the highest confluence. As curcumin concentration increases, this beneficial effect reduces, resulting in a lower confluence percentage. This trend is similar to that obtained in [14], in which the authors analyzed the effect of curcumin on the extent of wound healing in early-passage human dermal fibroblasts. In their experimental results, a 0.25 μM dose had an extent of wound closure of approximately 1.5 (control being 1). Both release models achieved a confluence of 1.4 times the controls for the same dose in our results. The rest of our results showed similar effects to those described in [14] for both release models. Likewise, the authors of [46] conducted a proliferation assay on olfactory ensheathing cells and tested curcumin’s effect at doses of 0.1, 0.5, 1, 10, and 20 μM. Low concentrations were more effective in stimulating cell proliferation, although a concentration of 0.5 was the optimal, not necessarily the lowest. It was found that curcumin’s stimulatory effect decreases for concentrations greater than 10 μM, as shown in the present study.

The authors of [50] observed the same dose-dependent effect on mouse 3T3-L1 preadipocytes. The cell culture was treated with various curcumin concentrations for 24 h. Low concentrations had a positive effect, increasing cell proliferation, while high concentrations had an inhibitory effect. Interestingly, a concentration of 0.02 μM of curcumin had the most stimulating effect on cell proliferation and migration. Moreover, only doses equal to or greater than 50 μM decreased cell proliferation. Contrary to our results, a dose of 20 μM stimulated cell proliferation after 24 h, although to a lesser extent than lower concentrations. Similarly, curcumin stimulates the proliferation of neural progenitor cells [69], 0.5 μM being the most effective concentration, whereas doses greater than 1 μM have an inhibitory effect. Curcumin’s dose-dependent effect may differ between cell lines, thus resulting in a different therapeutic window.

At doses above 10 μM, there is an inhibition in migration and proliferation of fibroblasts, and apoptotic cell death occurs [14,46,47,48,49]. The cytotoxic effect of high doses, especially 10, 15, and 20 μM, has been studied in various types of cancer cells [49] to demonstrate curcumin’s possible antitumor properties. These results were successfully replicated, as shown in Figure 10. For doses greater than 10 μM, the inhibitory effect increases as the concentration increases, resulting in worse results than the controls. The reduced cell migration can be seen in Figure 9 for both models, as only a limited number of internal cells have reached the scratch line. However, apoptotic cell death was not integrated into the present model to simplify the simulation’s output. Thus, this process could be added to develop a more realistic model.

4. Conclusions

In the present work, the integration and development of two models in Morpheus—a 2D fibroblast culture growth model and a curcumin diffusion model—were successfully achieved. Cell proliferation was controlled by Gérard and Goldbeter’s cell cycle model [32], achieving a simple yet overall realistic representation of this biological process. Curcumin release was accomplished with two models, Korsmeyer–Peppas and Peleg’s hyperbolic, using the experimental values reported in [17], with a release ratio of 0.91. Notably, a scratch-wound assay was recreated to simulate curcumin’s stimulative and inhibitory dose-dependent effects on wound healing, as described in numerous studies. The model was tested with six key concentrations—0.25, 0.5, 1, 5, 10, and 20 μM—and the final confluence of 447 initial seeded cells was compared to a control group.

Our results were similar to those described in experimental reports, especially in vitro scratch-wound assays. The stimulative effect of low curcumin doses was successfully recreated as cell migration, and a higher confluence than the control was obtained. The inhibitory effect for doses higher than 10 μM was achieved to reduce migration and cell proliferation. Thus, this model could be used for research into curcumin’s wound-healing effects as an auxiliary tool for experimental work. However, cell apoptosis was not implemented. This significant effect could be considered for future work to generate a more representative model.

Similarly, the parameters could be adjusted to the fibroblast growth model by implementing an in vitro fibroblast proliferation assay and using the experimental results as the initial values. Likewise, conducting in vitro scratch-wound assays treated with different curcumin doses and evaluating their effects can help to increase understanding and define curcumin therapeutic doses. Regarding the integrated model, some processes could be automatized, e.g., in the current model, the probability of p₁ and p₂ must be manually changed depending on the simulated curcumin concentration.