EO was extracted from thyme (Thymus vulgaris L.) leaves obtained from Condimentos Naturales Tres Villas, Puebla, Mexico. Microwave-assisted hydro-distillation was conducted using a NEOS microwave extraction system (800 W, 60 Hz) from Milestone, Sorisole, Italy. For the extraction process, a sample of 200 g of thyme was mixed with distilled water in a beaker, maintaining a solid-to-water ratio of 1:3 (Lainez-Cerón et al., 2022). The beaker was secured to the microwave-assisted extraction system, and the extraction process was carried out for 40 min at a stirring speed of 400 rpm.

EO was analyzed using the methodology described by Paris et al. (2020) with a gas chromatograph (Agilent 6850 N, Agilent Technologies, Santa Clara, CA, USA) coupled with a mass spectrometer (Agilent 5975 C). A 30 m × 0.25 mm column with a 0.25-μm film thickness (Agilent HP-5) was employed, using helium as the carrier gas at a 1.1 mL/min flow rate. The injector temperature was set to 250°C. The oven temperature started at 60°C for 2 min, then increased gradually to 10°C per minute until it reached 250°C. This temperature was maintained for an additional 10 min before rising to 300°C. Thyme EO volatile compounds were identified using the National Institute of Standards and Technology (NIST) library.

Aspergillus flavus and Penicillium citrinum were obtained from the collection of the Food Microbiology laboratory of the University of the Americas Puebla. Mold spores were harvested from 7-day slants using 5 mL of aqueous Tween 80 solution (0.1% v/v). The concentration of spores in the suspension was determined using a Neubauer chamber, and the spore suspension was adjusted to 1 × 10⁶ spores/mL.

A full-factorial experimental design was utilized to evaluate the effect of selected factors on the lag time, radial growth, and the growth or no-growth response of A. flavus and P. citrinum. Studied factors were a_w, thyme EO concentration, and pH. Solid model systems were prepared using potato-dextrose agar (PDA) as a base for mold growth. In each case, the culture medium was adjusted to different a_w values (0.99, 0.95, or 0.90) using sodium chloride. The amount of sodium chloride was calculated using the Favetto and Chirife (1985) Equation 1 as follows:

Where: K^* = 0.03710, constant for sodium chloride, and m = solute molality.

After a_w of PDA was adjusted, it was sterilized at 121°C for 15 min, cooled, and pH adjusted (5.0, 4.0, and 3.0) with filter-sterilized citric acid solution (10%). The acid amounts required to achieve each pH value were determined through titration curves. The PDA prepared for each condition was divided to add the quantities of thyme EO to reach the concentrations to be evaluated in each case (0, 25, 50, and 500 ppm), mixed, and poured into 60 × 15 mm Petri dishes until solidified. 5 μL of spore suspension was inoculated in the center of the agar and allowed to dry for 10 min under aseptic conditions. Inoculated plates and corresponding controls were incubated at 25°C within airtight plastic chambers to prevent anoxic conditions and were regularly inspected for fungal growth for 90 days (2,160 h). Colony diameters were frequently quantified using a digital caliper (Mitutoyo Corp., Kawasaki, Japan), with measurements taken along two orthogonal axes to ensure accuracy. Negative controls without EO were included in the experimental design. All systems were evaluated in triplicate.

In addition, each medium formulation’s pH and a_w, plates without fungal inoculation, were measured initially and after 90 days of incubation to verify that the pH and a_w conditions were maintained throughout the incubation period, ensuring that the observed delay in growth rate was not attributable to changes in these two factors. pH was measured using a potentiometer (model 50, Beckman, Brea, CA, USA) that had been previously calibrated. Water activity was assessed with a hygrometer (AQUALAB, series 3B, v.3.0, Decagon, Pullman, WA, USA) that had also been calibrated. All measurements were taken in triplicate.

The major compounds in thyme EO were p-cymene, thymol, carvacrol, caryophyllene, and γ-terpinene (Table 1), representing 90.4% of the total. Etri and Pluhár (2024) and Pluhár et al. (2024) noted that thymol in thyme EO could vary considerably, influenced by chemotypes, variety, region, and environmental factors. Najar et al. (2021) analyzed different chemotypes of thyme, the ‘thymol’ chemotype EO showed thymol as the major constituent (51.26–49.87%) followed by γ-terpinene and p-cymene, and in the ‘linalool’ chemotype, the EO contains oxygenated monoterpenes such as linalool (75%), linalyl acetate (8.15%), and b-caryophyllene (3.2%) as main constituents. The studied thyme variety contains trace amounts of copaene and camphene, each around 1.0%.

The Gompertz model is the sigmoidal representation of microbial growth kinetics, capturing key parameters such as maximum growth (A), specific growth rate (μ), and lag phase duration (λ). The Gompertz model is often used to analyze mold growth kinetics under varying environmental conditions (Figure 1); it has been employed to evaluate how changes in moisture and temperature affect mold growth; also, the model’s parameters offer valuable insights into mold growth rates and inhibition, necessary for assessing spoilage risk (Pitt and Hocking, 2022). Nielsen and Rios (2000) applied this model to study the effects of EOs on mold inhibition in bakery products, demonstrating how different EO concentrations influenced growth rates and lag phase periods. This highlights the model’s effectiveness in evaluating natural antifungal agents.

Table 2 presents the parameters of the Gompertz model, where the maximum growth, the growth rate, and the lag phase are observed. The control systems without EO showed that for a_w 0.99 at pH levels of 3, 4, or 5, mold began to grow after 25–27 h. P. citrinum and A. flavus exhibited higher growth rates at the highest a_w (0.99). In cultured media formulated with a_w 0.99 and pH 5, the growth rate is higher than at pH 3. In contrast, mold growth occurs later at a_w 0.95 and pH of 3, starting after 140 h, compared to pH levels of 4 or 5, where growth begins after 26 or 62 h, respectively. Growth decreased significantly as a_w was reduced, especially at a_w = 0.90, where both molds showed minimal or no growth across pH levels and EO concentrations tested (Table 2). At a_w of 0.90, mold growth occurs significantly later at a pH = 3, with a lag phase of more than 1,000 h. Reducing a_w to 0.99, 0.95, or 0.90 at pH levels 3, 4, or 5 does not effectively inhibit mold growth without thyme EO. Adding 25 ppm of thyme EO influences the mold response (Figure 1). The lag phase is higher at pH 3 than at pH 5 for concentrations of 0, 25, and 50 ppm of EO. For concentrations of 100 ppm, the growth rate is lower at pH 3 than at pH 5. Figure 1 shows that the Gompertz model fits the experimental data well; the R² was >0.90 in the analyzed cases.

Polynomial models (Table 3) may be used to predict the mold’s behavior under different factor combinations. Figure 2 shows the surface response graphs for the studied molds’ lag time (h) under different combinations of a_w and thyme EO in media formulated at pH 4.0. P. citrinum growth remained relatively stable across pH levels when other conditions favored growth, while A. flavus showed variable growth rates and lag phases with changing pH. Acidic pH (around pH 3) tended to reduce growth rates and increase lag phases in both molds, indicating pH as a limiting factor for mold proliferation. Similar findings have been reported previously for Penicillium expansum (Tannous et al., 2016) and A. flavus (Kosegarten et al., 2017). The lag phase typically increases as a_w decreases, showing a delay in mold growth (Table 2; Figure 2). Dagnas et al. (2014) also recorded increasing lag times at decreasing a_w for A. niger and Penicillium corylophilum in malt extract agar. Growth rates were generally higher at pH 5 for both molds, although pH sensitivity was more pronounced in A. flavus than in P. citrinum as previously mentioned. Both molds demonstrated considerable sensitivity to increasing thyme EO concentrations (Figure 2). At ≥50 ppm, P. citrinum growth was almost completely inhibited (indicating it is more sensitive to this factor), while A. flavus could still grow but with a substantially longer lag phase and reduced growth rate. EO concentrations of 100 or 500 ppm inhibited the growth of both molds entirely (Table 2). This indicates a concentration-dependent inhibition for both molds, but A. flavus seems to tolerate slightly higher concentrations than P. citrinum before complete growth inhibition occurs. A. flavus showed slightly higher resistance to lower EO concentrations than P. citrinum, maintaining growth at 25 and 50 ppm in certain conditions (Figure 2). Under subinhibitory conditions (low pH, high EO concentration), A. flavus generally displayed longer lag phases than P. citrinum, suggesting it may take longer to adapt to challenging environments.

The findings reported align with those from authors who evaluated the antifungal effects of thyme EO or its components in various combinations with other factors. Suhr and Nielsen (2003) demonstrated that thymol exhibits greater inhibitory effects at pH 5.5 compared to pH 6.5. At acidic pH, thyme EO or thymol molecules become less dissociated and more hydrophobic, enhancing their ability to bind to hydrophobic regions of proteins and dissolve in the lipid phase (Oliveira et al., 2020; Ranjbar et al., 2022). This possibly explains the disruption of membrane permeability (Oliveira et al., 2020; Ranjbar et al., 2022), confirmed by the efflux of intracellular K⁺ ions (Hu et al., 2018). Similarly, Thompson (1990) observed that 1.0 mM carvacrol completely inhibited the growth of A. flavus at pH 4 and 8. Mycelial growth was partially inhibited at lower concentrations (0.1 and 0.5 mM), with the most significant reductions at these pH levels. Schlösser and Prange (2019) evaluated the impact of carvacrol on the mycelial growth of Penicillium verrucosum and Aspergillus westerdijkiae. Their findings revealed that supplementation with ½ the minimum inhibitory concentration (MIC) of carvacrol significantly reduced the mycelial dry weight of both molds throughout a 21-day incubation period. Debonne et al. (2019) reported that the growth of Penicillium paneum was significantly reduced at a_w 0.88 compared to higher a_w levels (0.93–0.97). Additionally, the lowest concentration of thyme EO required to inhibit growth over 144 h was 1 μL/mL. The evaluation of the antifungal activity of thyme EO in vitro against A. niger and P. paneum confirmed that thyme EO at concentrations between 0 and 0.5 μL/mL significantly (p < 0.05) influenced both λ and μ_max under various temperature conditions (Debonne et al., 2018).

Over the last four decades, predictive models originally developed for bacteria have been adapted to describe mold growth, including both primary and secondary models (Dagnas and Membré, 2013). These models allow food manufacturers to design product formulations that achieve a targeted shelf life or, conversely, to adjust shelf life based on the formulation. The binary logistic regression model has proven effective in determining the conditions under which mold growth occurs or is inhibited (Hernández-Figueroa et al., 2024; Kosegarten et al., 2017). The binary logistic regression model is a widely used statistical tool for predicting the probability of a binary outcome, such as the presence or absence of mold growth under specific environmental conditions. This model categorizes conditions as favorable or inhibitory to microbial growth, thereby allowing the assessment of key environmental factors such as pH, a_w, and the antimicrobial agents concentration (Henley-Smith et al., 2014; Possas et al., 2021). However, the binary logistic model does not quantify the extent or rate of mold growth; it merely identifies whether conditions favor growth.

This study employed binary logistic regression to predict the probability of A. flavus and P. citrinum growth as a function of a_w, pH, and EO concentration. The results indicated that a_w and pH were significant predictors, with higher a_w levels positively correlated with fungal growth, consistent with the established role of water availability in supporting microbial activity. Similarly, more neutral pH values (closer to pH 5) were associated with increased growth probability. Conversely, higher EO concentrations exhibited a significant inhibitory effect on fungal growth. These findings align with prior research demonstrating that antimicrobial compounds, such as EOs, can effectively suppress fungal activity (Al-Maqtari et al., 2021).

The interaction between factors provided additional insights (Table 4; Figure 3). For instance, a_w and thyme EO concentration interaction affects the probability of mold growth (Figure 3), and the inhibitory effect of EO was more pronounced under conditions of lower a_w, suggesting a synergistic potential for combining these variables in mold control strategies (Figure 3). Likewise, binary interactions of factors a_w-cinnamon EO, a_w-pH, and other combinations of fat content, protein content, temperature, and pH impacted A. flavus growth probability in PDA systems (Kosegarten et al., 2017). In another study, binary interactions of time, type of edible film application, and EO concentration determined the growth probability of Penicillium commune, Cladosporium herbarum, and Fusarium spp. on tomatoes (Hernández-Figueroa et al., 2024). According to the predictive model (Table 4), several conditions decrease the growth probability of P. citrinum to values below 0.05. These conditions include an a_w of 0.95 at a pH of 3, combined with 20–25 ppm of EO, or an a_w of 0.94 at a pH of 5 and 35 ppm, as shown in Figure 3. For a_w lower than 0.93 combined with a pH of 3.0, EO at <10 ppm achieves this growth probability. These conditions balance the need for a low probability of fungal growth with practical constraints in food formulation (lower EO concentration). Based on the binary logistic regression, the developed predictive framework was designed to understand the conditions that favor or inhibit A. flavus and P. citrinum growth under varying a_w, pH, and EO concentration levels (Figure 3). The binary logistic regression model revealed important findings: higher a_w increases fungal growth, while higher concentrations of EO inhibit it.

Time-to-growth regression is an important method for analyzing time-to-event data, commonly used in survival analysis and shelf-life estimation for food products (Bandyopadhaya et al., 2024; González-Pérez et al., 2024). In food preservation, this method helps determine the length of the lag phase—the period before mold growth begins—under different environmental conditions. By modeling the effects of factors such as a_w, pH, and EO concentrations, this procedure provides insights into how these variables can delay the onset of mold growth. However, it is important to note that while this method effectively quantifies growth delays, it does not describe the subsequent growth rate or the extent of mold proliferation after the lag phase (Zaffora et al., 2024; Dantigny et al., 2005; Gougouli et al., 2011). Survival analysis is valuable for examining the lag phase before mold growth, including under antifungal agents. This method helps researchers measure the time it takes for molds like Aspergillus to grow, providing insights for shelf-life extension strategies (Gómez-Ramírez et al., 2013; Kosegarten et al., 2017). Suhr and Nielsen (2004) evaluated weak acid preservatives’ effect on the growth of spoilage fungi at different a_w and pH values. The obtained data was modeled using survival analysis to determine ‘spoilage-free time’ for the fungi. At the low a_w level (0.80), only Eurotium species grew within the test period of 30 days. Higher a_w levels and higher pH values decreased the spoilage-free times.

The time-to-growth regression method effectively quantifies conditions that delay the growth of A. flavus and P. citrinum (Figure 4). As can be seen, depending on the selected time (100, 500, or 1,500 h) to delay the growth of each mold studied, different a_w and EO concentrations at pH 5 are required. In the same way, predictions of the time to growth can be made at other pHs (Table 4). In this study, a_w exhibited a negative coefficient (Table 4), indicating that higher a_w levels reduce the time required for growth to begin. Similarly, pH showed a negative effect, suggesting that shifting toward neutral pH accelerates mold growth. Conversely, the thyme EO concentration had a strong positive impact, significantly delaying mold growth at higher concentrations. Interaction terms showed the combined effects of the studied variables (Table 4). The interactions a_w-pH, pH-EO concentration, and a_w-EO concentration revealed synergistic or antagonistic effects on the mold’s time to growth. The regression model also offers practical recommendations for maximizing the mold’s lag phase under specific constraints. These findings emphasize the importance of considering combined factors rather than individual variables in preservation strategies (López-Malo et al., 2005; Mannaa and Kim, 2017; Samapundo et al., 2007). The regression model also offers practical recommendations for maximizing the lag phase of P. citrinum under specific constraints. For conditions with a_w 0.95 and minimal thyme EO concentration (25 ppm), pH 5 (Figure 4) can extend the lag phase and delay growth detection. Predictions indicate that these conditions could inhibit P. citrinum growth for over 1,000 h (~42 days). A shorter lag time (20 days) had been reported for A. niger growth at a_w 0.90 and <50 ppm of Mexican oregano EO in dried tomatoes (Gómez-Ramírez et al., 2013).

Based on the results of the binary logistic regression and regression with life data analyses, the growth responses of A. flavus and P. citrinum can be compared under varying a_w, pH, and EO concentration conditions. For A. flavus, the logistic regression analysis indicated a strong positive effect of a_w on fungal growth probability, with high a_w levels significantly increasing the probability of growth. In the regression with the life data model, higher a_w led to faster growth onset. In contrast, P. citrinum was slightly less sensitive to changes in a_w. Although increased a_w did raise the probability of growth for P. citrinum, the magnitude of the effect was less pronounced than for A. flavus. For A. flavus, pH showed only a marginal impact on both models. A lower pH (around 3–4) slightly reduced the growth probability and delayed the growth onset, but this effect was not as pronounced as the effects of a_w and EO concentration. In the case of P. citrinum, pH had a more substantial influence on growth probability and time-to-growth. Marin et al. (2021) highlighted that numerous studies have been published in recent years focusing on applying predictive mycology for the early prediction of fungal spoilage. Examining the lag time before fungal colonies appear is crucial, as molds are undesirable in several food products. Many studies use a growth/no-growth approach to define the boundaries influenced by various factors. This shift in predictive mycology—focusing on lag times and the probability of colony appearance—represents a significant advancement. While the kinetic model can fully describe growth dynamics. Together, these models would give a holistic understanding of how environmental factors impact mold growth, particularly useful in food safety and shelf-life studies. Dantigny (2021) highlights that fundamental principles from process control can be applicable to process modeling, emphasizing that the simplest model that effectively represents the data is the most desirable. A thorough understanding of the process is essential since no model can encompass all the complexities of a system. Identifying key biological responses and the significant factors influencing them is important since each model provides unique insights into mold growth.