3-D stochastic modeling approach in thermal inactivation: estimation of thermal survival kinetics of Escherichia coli O157:H7 in a hamburger after exposure to desiccation stress

ABSTRACT

Desiccation tolerance of pathogenic bacteria is one strategy for survival in harsh environments, which has been studied extensively. However, the subsequent survival behavior of desiccation-stressed bacterial pathogens has not been clarified in detail. Herein, we demonstrated that the effect of desiccation stress on the thermotolerance of Escherichia coli O157:H7 in ground beef was limited, and its thermotolerance did not increase. E. coli O157:H7 was inoculated into a ground beef hamburger after exposure to desiccation stress. We combined a bacterial inactivation model with a heat transfer model to predict the survival kinetics of desiccation-stressed E. coli O157:H7 in a hamburger. The survival models were developed using the Weibull model for two-dimensional pouched thin beef patties (ca. 1 mm), ignoring the temperature gradient in the sample, and a three-dimensional thick beef patty (ca. 10 mm), considering the temperature gradient in the sample. The two-dimensional (2-D) and three-dimensional (3-D) models were subjected to stochastic variations of the estimated Weibull parameters obtained from 1,000 replicated bootstrapping based on isothermal experimental observations as uncertainties. Furthermore, the 3-D model incorporated temperature gradients in the sample calculated using the finite element method. The accuracies of both models were validated via experimental observations under non-isothermal conditions using 100 predictive simulations. The root mean squared errors in the log survival ratio of the 2-D and 3-D models for 100 simulations were 0.25–0.53 and 0.32–2.08, respectively, regardless of the desiccation stress duration (24 or 72 h). The developed approach will be useful for setting appropriate process control measures and quantitatively assessing food safety levels.

IMPORTANCE

Acquisition of desiccation stress tolerance in bacterial pathogens might increase thermotolerance as well and increase the risk of foodborne illnesses. If a desiccation-stressed pathogen enters a kneaded food product via cross-contamination from a food-contact surface and/or utensils, proper estimation of the internal temperature changes in the kneaded food during thermal processing is indispensable for predicting the survival kinetics of desiccation-stressed bacterial cells. Various survival kinetics prediction models that consider the uncertainty or variability of pathogenic bacteria during thermal processing have been developed. Furthermore, heat transfer processes in solid food can be estimated using finite element method software. The present study demonstrated that combining a heat transfer model with a bacterial inactivation model can predict the survival kinetics of desiccation-stressed bacteria in a ground meat sample, corresponding to the temperature gradient in a solid sample during thermal processing. Combining both modeling procedures would enable the estimation of appropriate bacterial survival kinetics in solid food.

INTRODUCTION

Bacterial cells are often exposed to environmental stresses, such as desiccation and poor nutrition in a food processing facility and/or home kitchen. Changes in bacterial properties and population dynamics may occur through exposure to environmental stress. Changes in inactivation kinetics after stress exposure have also been previously reported (1–3). These studies show that stress-exposed bacterial cells may exhibit different survival kinetics from cells not exposed to stress. However, few studies have been conducted on the inactivation kinetics of desiccation-stressed pathogenic bacteria in food products because of the complexity of the experimental setups needed. Appropriate estimation of the inactivation kinetics of pathogenic bacteria on food surfaces after desiccation stressed through cross-contamination will contribute to the accurate estimation of quantitative microbial risks and ensure microbiological safety.

Microbiological prediction models play an important role in microbial risk assessment (4–6). Mathematical model-based predictions have been used to estimate the number of bacteria in food products (7–14). However, models that predict the bacterial inactivation kinetics after exposure to desiccation stress have not yet been developed. Furthermore, variations in bacterial kinetics occur because of the variability of individual cells and the uncertainty from experiments and regressions performed. Because the environmental stresses such as desiccation and/or starvation vary depending on the contamination route, duration, temperature, and humidity of the space, greater variations in survival/growth kinetics could be observed because of these factors. Therefore, stochastic models must be used to consider variations when predicting bacterial kinetics after exposure to desiccation stress.

A kind of cross-contamination is likely to occur during the manufacturing of ground/minced meat owing to the nature of the grinding machine used. Since cross-contaminated bacterial cells can be easily mixed inside a kneaded meat product, such as a hamburger, an appropriate prediction of the number of surviving bacteria corresponding to the internal temperature change is needed to ensure the microbial safety of the food product. Recent attempts have been made to simulate heat transfer processes in foods using finite element method (FEM) software (15–19). These studies suggest that the temperature change could be estimated based on heat transfer models. However, because heat transfer models alone cannot predict bacterial inactivation kinetics, combining a heat transfer model with a bacterial inactivation model will allow the prediction of the survival kinetics of cross-contaminated bacteria in a ground meat sample.

The thermotolerance of desiccation-stressed pathogenic bacteria in real food has not yet been investigated in detail. Determining the thermotolerance of desiccation-stressed pathogenic bacteria would ensure the appropriate process control measures are provided. Furthermore, developing a predictive model for the survival of desiccation-stressed pathogenic bacteria will enable the quantitative evaluation of food safety levels. The objectives of this study were to quantify the survival kinetics of Escherichia coli O157:H7 in a beef hamburger after exposure to desiccation stress on stainless steel surfaces and to predict the survival kinetics of E. coli O157:H7 in a beef hamburger corresponding to the temperature change during heating using a heat transfer model.

RESULTS

Inactivation kinetics of E. coli O157:H7 during isothermal heating after direct contamination and cross-contamination

The survival kinetics of E. coli O157:H7 with or without desiccation stress in ground beef were examined at 55°C, 57.5°C, 60°C, and 62.5°C (Fig. 1), and the Weibull parameters were estimated with those for the goodness-of-fit (root-mean-square-error, RMSE) for each condition (Table 1). As the initial cell numbers changed owing to desiccation stress on the stainless-steel surface, the inactivation behavior of E. coli O157:H7 inoculated in ground meat was evaluated using log₁₀ survival ratios (log N/N₀). Both contamination pathways with or without desiccation stress showed almost the same inactivation kinetics at 55°C and 57.5°C (Fig. 1a), indicating similar values for the Weibull rate parameter δ (Table 1). Apparent differences in the survival kinetics were observed between the two different contamination pathways at 60°C and 62.5°C (Fig. 1b). The desiccation-stressed E. coli O157:H7 in ground meat was inactivated more rapidly than those directly contaminated, which was reflected as a lower δ value for the cross-contamination pathway than that of direct contamination (Table 1).

Fig 1.

Inactivation kinetics (log N/N₀) of 24-h-desiccation-stressed (closed symbols) or stress-free (open symbols) E. coli O157:H7 in ground beef under isothermal conditions at relatively low temperature (a) 55°C (●, ○), 57.5°C (▲, ), and relatively high temperature (b) 60°C (◆, ◇), and 62.5°C (■, □). The results are presented as the mean ± standard deviations of triplicate trials, and the fitted lines are described using the modified Weibull model. The log N₀ was 7.5 ± 0.1 log CFU/g. × indicates that no colonies were detected (detection limit: 100 CFU/g).

TABLE 1.

Weibull parameters of stressed for 24 h or stress-free E. coli O157:H7^a

Temperature (°C)	Conditions	δ (s)	p (-)
55	Stressed	1,391 ± 626	1.08 ± 0.366
	Stress-free	1,178 ± 178	0.990 ± 0.183
57.5	Stressed	263 ± 49.7	0.792 ± 0.146
	Stress-free	212 ± 52.3	0.778 ± 0.116
60	Stressed	53.3b ±6.93	0.901 ± 0.111
	Stress-free	109b ±8.21	1.11 ± 0.189
62.5	Stressed	21.7b ±0.37	1.33 ± 0.0759
	Stress-free	36.3b ±3.58	1.23 ± 0.189

^a Each value is expressed as mean ± standard deviation of triplicate trials.

^b Means are significantly different (P < 0.05).

Validation of the 2-D model in non-isothermal heating conditions

The parameter distributions of δ and p derived from the bootstrap were convergent because the values of R-hut were 1.0 at all temperature conditions. A log-linear regression for δ on temperature was performed on a set of 1,000 Weibull parameters obtained via Weibull fitting to a resampled data set obtained from bootstrapping (Fig. 2a). The log δ values showed a linear decrease with increasing temperature. In contrast, the scale parameter p was almost constant regardless of the heating temperature (Fig. 2b).

Fig 2.

Changes in the log₁₀δ (s) (a) and p (-) (b) values as functions of heating temperatures (data points derived from the condition without desiccation stress). Error bars indicate the standard deviation of 1,000 estimated Weibullian parameter replicates. Lines indicate the secondary model derived from 1,000 bootstrap replicates. The parameters a, b, and c estimated averages with 95 percentiles in parentheses for the secondary model of 𝑙𝑜𝑔₁₀𝛿 = 𝑎 × 𝑇 + 𝑏 and 𝑝 = c were −0.19 (-0.20,–0.18), 13.65 (12.86, 14.32), and 1.04 (0.92, 1.15), respectively.

Figure 3 compares the predicted inactivation kinetics of E. coli O157:H7 exposed to different desiccation stresses inoculated into a ground beef hamburger during non-isothermal heating (as shown in Fig. 8) with the observed experimental data. The survival kinetics of E. coli O157:H7 were almost the same regardless of the level of desiccation stress. The average RMSE values for 100 simulations were 0.338, 0.249, and 0.534 for direct contamination (Fig. 3a), 24-h dry stress cross-contamination (Fig. 3b), and 72-h dry stress cross-contamination (Fig. 3c), respectively.

Fig 3.

Inactivation kinetics of E. coli O157:H7 with or without desiccation stress in 5 g of ground beef with a thickness of 1.0 mm. The probabilistic simulation (gray 1,000 lines) and observed values without desiccation stress (○) (a), desiccation-stress-24 h (△) (b), and desiccation-stress-72 h (□) (c) are shown. The dotted horizontal lines represent the detection limit. × indicates that no colonies were detected.

Validation of the 3-D model

We predicted the temperature variation in the hamburgers with a three-dimensional (3-D) structure using the FEM and then used the temperature history to predict the variation in the number of surviving E. coli O157:H7 cells. Figure 4a shows a comparison of the FEM predictions with the observed temperatures measured every minute at the center of the hamburger during the heating process. The residuals between the predicted and observed temperatures were large in the initial phase (<10 min) but decreased as heating proceeded (Fig. 4b). The temperature gradient during heating, as estimated using COMSOL, is shown in Fig. 5. A video of the predicted changes in the temperature gradient during heating is provided in the supplemental material (Fig. S1).

Fig 4.

Comparison of the predicted (solid line) and measured (plot) center temperatures of the hamburgers (a) and the corresponding residuals (b). The actual center temperatures were measured every minute.

Fig 5.

Predicted temperature gradient inside the hamburger (t = 10 min). The color bar on the right side represents the temperature (°C).

Figure 6 compares the predicted inactivation kinetics of E. coli O157:H7 in the whole hamburger patties with the observed data from the validation experiments. The average RMSE values for 100 simulations were 2.08, 0.696, and 0.324 for direct contamination (Fig. 6a), 24-h dry stress cross-contamination (Fig. 6b), and 72-h dry stress cross-contamination (Fig. 6c), respectively. However, there was a large discrepancy between the observed and predicted values in the 20.8 min (1,250 second) survival ratio of the directly contaminated 3-D samples (Fig. 6a). Based on the temperature gradient during the heating process (Fig. 5), the survival ratio gradient during heating, as estimated using COMSOL, is shown in Fig. 7. The survival ratio corresponded well to the temperature gradient. To clearly show the survival gradient during the heating process, a video is provided in the supplemental material (Fig. S2).

Fig 6.

Inactivation kinetics of E. coli O157:H7 with or without desiccation stress in 100 g of beef hamburgers with a thickness of 1.7 cm. The probabilistic simulation (gray 1,000 lines) and observed values without desiccation stress (○) (a), desiccation-stress-24 h (△) (b), and desiccation-stress-72 h (□) (c) are shown. The dotted horizontal lines represent the detection limit. × indicates that no colonies were detected.

Fig 7.

Predicted survival ratio gradient inside the hamburger (t = 10 min). The color bar on the right side represents the survival ratio (log N/N₀).

DISCUSSION

Environmental stress during cross-contamination did not increase the thermostability of E. coli O157:H7, while higher heating temperatures tended to decrease the thermostability (Fig. 1). Drying stress may have weakened their resistance to high-temperature treatment. Shaker et al. (20) reported the inactivation kinetics of Enterobacter sakazakii in infant formula after exposure to environmental stress (20). They reported that drought stress significantly reduced the D-values, which is similar to the results of this study using E. coli O157:H7, which belongs to the same order as E. sakazakii (Enterobacterales). However, the route of bacterial contamination cannot be determined until food poisoning occurs. Therefore, the predictive modeling of bacterial inactivation kinetics should be tailored to a more thermotolerant route of bacterial contamination: direct contamination.

The stochastic two-dimensional (2-D) model developed in this study predicted the inactivation kinetics of E. coli O157:H7 in ground beef after both direct and cross-contamination (Fig. 3). The results indicate that stochastic nonlinear modeling is effective in predicting inactivation kinetics under a dynamic temperature history based on experimental data obtained under constant temperature conditions. One of the most popular models is the log-linear approach based on the decimal reduction time (D-value), which is used to calculate the inactivation of E. coli O157:H7 in ground beef (7, 10, 11). However, the validity of the D-value has been questioned in some studies (21, 22). Van Boekel (23) showed that first-order kinetics is an exception rather than the rule upon heat treatment, suggesting that this concept may lead to an overestimation or underestimation of the thermal deactivation time (23). Additionally, it has been noted that deterministic predictions involving the D-value concept do not account for uncertainty (24, 25). Uncertainty arises from a lack of knowledge regarding the correct parameters and includes measurement and regression errors (26). Many researchers have argued the importance of a probabilistic approach in quantitative microbial risk assessment (6, 27–29) and have attempted to stochastically predict microbial inactivation (26, 30–33). The method used in this study enables appropriate predictions over variable temperature histories by solving the modeling issues of the deterministic and log-linear approaches.

The 3-D model used in this study predicted the inactivation kinetics of E. coli O157:H7 in a hamburger with a temperature gradient (Fig. 6), which was supported by heat transfer simulations using COMSOL that were able to predict the center temperature with high accuracy. Although large residuals in the center temperature were observed in the early stage of heating, they decreased in the later stage of heating when bacterial inactivation progressed. Modeling the heat transfer process inside food during heating has demonstrated its usefulness in predicting bacterial behavior. Most thermal inactivation models can predict non-isothermal conditions, assuming the volumetric sample size is small. As the sample size increases, a spatial temperature gradient is introduced, resulting in heterogeneous bacterial dynamics within the sample. Some studies have attempted to predict the bacterial kinetic behavior using heat transfer simulations (34, 35). However, these studies used the D-value approach for the primary model, and their predictions were based on a deterministic approach. In contrast, the 3-D model developed in this study has three characteristics: a nonlinear approach, stochastic prediction, and heat transfer simulation. This modeling procedure contributes to the development of practical microbiological prediction models for future applications.

The differences in the survival kinetics between the contamination routes under non-isothermal conditions were smaller than those observed under isothermal conditions (Fig. 1b, 2b, and c). An underestimation of the surviving bacterial counts was also observed in the directly contaminated 3-D samples (Fig. 6a). There are several possible reasons for these differences, one of which is the variability in bacterial inactivation dynamics. Low bacterial counts increase variability in kinetic behavior owing to bacterial heterogeneity (22, 30, 36). Because the surviving bacterial counts after 8 min under non-isothermal temperature conditions in 2-D samples and at 20.8 min in 3-D samples were close to the detection limit (200 CFU/sample), the discrepancy might become large due to increased variability. These variations in bacterial counts are inevitable when evaluating bacterial survival kinetics.

This study has some limitations. First, the developed model was for ground beef; however, when applied to other foods, the heat transfer simulation only requires changing the geometry and composition of the ingredients. Previous studies attempted to quantify or model the degree to which bacteria are transferred during cross-contamination (37–39). Applying the results of these studies to the initial bacterial counts in this model would allow more realistic predictions. Although only temperature changes were simulated using COMSOL in this study, the prediction accuracy of the model could be improved by calculating phenomena other than heat transfer. For example, myosin, a type of myofibrillar protein, denatures at approximately 50°C in beef (40), resulting in weight loss and volume change. Shape changes and water loss may also need to be considered when simulating more complex foods in the future.

In conclusion, this study developed an experimental procedure to reproduce cross-contamination and found that the environmental stress of cross-contamination did not increase the thermotolerance of E. coli O157:H7 cells. Moreover, the developed model could predict the kinetic behavior of cross-contaminated E. coli O157:H7 with high accuracy. In addition, the 3-D stochastic prediction method used in this study demonstrated that it can represent the uncertainty in bacterial kinetic behavior while accurately capturing the temperature gradient inside food. Collectively, these results will be useful for a more realistic risk assessment considering the effects of cross-contamination.

MATERIALS AND METHODS

Bacterial strains

A cocktail of four strains of enterohemorrhagic E. coli O157:H7 [HIPH 12361, kindly provided by the Hokkaido Institute of Public Health; RIMD 05091896, RIMD 05091897, and RIMD 0509939 were obtained from the Research Institute for Microbial Diseases (RIMD), Osaka University] was used for all experiments. All strains were maintained at −80°C in tryptic soy broth (TSB; Merck, Darmstadt, Germany) containing 10% glycerol. Using a platinum loop, the frozen cultures were transferred to trypticase soy agar plates (Merck), which were incubated at 37°C for 24 h. An isolated typical colony of each strain was transferred to 5 mL of TSB (Merck) independently in a plastic tube and incubated at 37°C for 24 h. Next, 100 µL of the bacterial culture was transferred to 5 mL of fresh TSB and incubated for another 24 h. Afterward, the bacterial cells were collected via centrifugation (3,000 × g for 10 min at 25°C). The resulting pellet of each strain was washed twice with sterile water and resuspended in 1 mL of sterile water. Finally, each strain suspension (1 mL) was added before the experiments.

Sample preparation for E. coli O157:H7 inactivation in ground beef samples

Cross-contaminated and directly contaminated ground beef were used as experimental samples in this study. Beef thigh chunks were fed into a sterile electric mincer to produce ground beef. A 2-mL aliquot of the bacterial suspension (9.6 log CFU/mL) was placed onto the surface of each stainless steel bowl in tiny droplets (50 µL). The contaminated bowl was dried in an electric dryer at 25°C and 50% relative humidity (RH) for 3 h. The contaminated stainless steel bowls were then stored at 15°C and 50%–60% RH for 24 and 72 h to simulate desiccation stress. Ground beef thigh (ca. 100 g lean meat) was kneaded in a contaminated stainless steel bowl to simulate cross-contamination. A small portion of the contaminated ground beef (5 g) was placed in a stomacher bag and stretched to a thickness of 1 mm. The thin-layer beef samples were sealed using a heat sealer (PC-200; Fuji Impulse, Osaka, Japan); these samples were defined as the “with desiccation-stressed” group. For comparison, a “without desiccation-stressed” group was also prepared. Briefly, 100 g of ground beef and a 1.0-mL aliquot of the bacterial suspension (9.6 log CFU/mL) were simultaneously placed in an uncontaminated bowl and kneaded thoroughly for 1 min. The ground beef was placed in stomacher bags in the same way as described above.

Isothermal treatment for model development

The sealed thin-layer ground beef pouches were immersed in a water bath (TR-2α; AS ONE, Osaka, Japan) in four isothermal conditions: 55°C, 57.5°C, 60°C, and 62.5°C for 80, 20, 4, and 1.33 min, respectively. The temperature profile of the thin-layer ground beef was confirmed in advance using K-type thermocouples; the come-up time was not included in the heating period. Five sealed thin-layer ground beef pouches were used for each temperature condition to determine the changes in the number of viable E. coli O157:H7 cells during each heating period at certain time intervals. When the thin-layer ground beef pouch was taken from the heating bath after a certain exposure period, the sample was immediately immersed in a cold water bath (<10°C) to remove excess heat. Samples were analyzed in duplicate at each sampling time and heating temperature.

Non-isothermal treatment for 2-D model validation

To validate the model by taking into account temperatures other than those used in the constant temperature condition, the contaminated thin-layer ground beef pouch was heated in a dynamic temperature profile, increasing the temperature at 1.1°C/min from 55°C to 62.5°C as shown in Fig. 8. This temperature profile imitated a Sous vide process to simplify the temperature change and heat transfer. Twenty sealed thin-layer ground beef pouches were used for the non-isothermal inactivation process to determine the changes in the number of viable E. coli O157:H7 cells at certain time intervals for 10 sampling times. Samples were tested in duplicate at each sampling time point.

Fig 8.

Non-isothermal heating protocol using a water bath. The samples were immediately chilled after heating.

Sample preparation for 3-D hamburgers

Cross-contaminated and directly contaminated beef patties were used as three-dimensional experimental samples. A 0.5 mL aliquot of the bacterial suspension was placed onto the surface of the stainless steel bowl in tiny droplets (50 µL). The contaminated bowls were dried as mentioned in the section “Validation of the 2-D model in non-isothermal heating conditions,” and they were stored at 15°C and 50%–60% RH for 24 h for bowls contaminated with 0.5 mL of the bacterial suspension and 72 h for bowls contaminated with 0.6 mL of the bacterial suspension. The amount of bacterial suspension used was determined such that the number of bacteria after drying would be the same while minimizing the effect on the moisture content of the food. Ground beef thigh (ca. 100 g) was then kneaded in a contaminated stainless steel bowl to simulate cross-contamination. The contaminated ground beef thigh was formed into patties with a radius of approximately 4.5 cm and a height of approximately 1.7 cm, avoiding the formation of voids. The beef samples were degassed in plastic bags and sealed using a vacuum sealer (Alice V952; Flaem Nuova, San Martino della Battaglia, Italy). These samples were defined as the “cross-contamination” group. For comparison, 100 g of ground beef and a 10 µL aliquot of the bacterial suspension were simultaneously placed in an uncontaminated bowl and kneaded thoroughly for 1 min. The beef was placed in stomacher bags as described above; these samples were defined as “directly contaminated.” Five sealed hamburgers were immersed in a water bath at 62.5°C to determine the changes in the number of viable E. coli O157:H7 five times during the heating period at certain time intervals. The temperature profile of the sample center was measured in triplicate using K-type thermocouples.

Enumeration of E. coli O157:H7

Whole ground beef samples in sealed pouches (5 g) were transferred to a stomacher bag and homogenized with 45 mL of 0.1% peptone water for 1 min using a stomacher. Whole hamburger samples (ca. 100 g) were also transferred to a stomacher bag and homogenized with 100 mL of 0.1% peptone water for 1 min using a stomacher. The homogenate (0.25 mL) was surface-plated in quadruplicate, and samples were serially 10-fold diluted in 0.1% peptone water (0.1 mL) and plated in duplicate onto CHROMagar O157:H7 plates (CHROMagar, Paris, France). The plates were incubated at 37°C for 24 h, and the number of colonies was counted. Each experiment was independently repeated three times for each temperature condition.

Estimation of bacterial survival kinetics in the 2-D model

Calculation of the Weibull parameter distribution

The model was developed based on a previous study (31) using the Weibull model as the primary model. The Weibull model is one of the most commonly used mathematical models to describe the inactivation behavior of bacteria. The Weibull equation was modified by Mafart et al. (41) as follows:

$${\displaystyle {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\displaystyle \frac{N\left(t\right)}{N_0}}={\mathrm{l}\mathrm{o}\mathrm{g}}_{10}S\left(t\right)=-{\left({\displaystyle \frac{t}{\delta }}\right)}^p}$$ (1)

where t is the treatment time (s) and N(t) and N₀ are the bacterial cell densities (CFU/g) at times t and 0, respectively. δ (scale parameter) and p (shape parameter) are the characteristic parameters of the modified Weibull model.

To obtain the parameter distribution of the Weibull model and consider the fitting uncertainties and thermotolerance heterogeneity of the individual cells, a sample set of 1,000 iterations was resampled for Weibull fitting using nonparametric bootstrapping. The bootstrap method (42) is a resampling technique used to estimate statistical parameters through computer simulations. Bootstrapping describes statistical parameters or values as a distribution, and it can represent their variation caused by the lack of information through random resampling following observed values. Survival data from three replicates were resampled thrice, including duplicates at each heating temperature. Each set of three samples was randomly selected as the resampled data from the set of direct contamination samples at all heating temperatures. Each resampled set was used as a bootstrapped data set to generate 1,000 sets.

To describe the relationship between the temperature and Weibull parameters, 1,000 sets of secondary model parameters were calculated. Based on the results of previous studies (23, 43), the temperature dependence of the Weibull model parameters δ and p were described. The secondary model of δ was then described as follows:

$${\displaystyle {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\delta =a\times T+b}$$ (2)

It has been reported that p(T) varies linearly or remains constant with increasing and decreasing temperatures (23). Hence, the secondary model of p was described as the average of the values at each temperature, as follows:

$$p=c$$ (3)

The Gelman–Rubin convergence statistic (R-hut value) was used to confirm the convergence of parameter distributions using bootstrapping. R-hut values < 1.1 indicate the convergence of the parameter distributions (44).

Prediction of survival kinetics in non-isothermal conditions

A prediction model under non-isothermal conditions was developed using the differential equation in equation (1) as follows (45):

$${\displaystyle {\displaystyle \frac{d}{dt}}\left[{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}S\right]=-\frac{p}{\delta \left(T\right)}{\left\{\delta \left(T\right){\left[-{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}S\left(t\right)\right]}^{{\displaystyle \frac{1}{p}}}\right\}}^{p-1}}$$ (4)

We randomly sampled δ(T) and p values for equation (4) from 1,000 sets of secondary models, and the predictions for S(t) were repeated 100 times to express the uncertainty in the inactivation kinetics. All statistical calculations were performed using the statistical environment R software (version 4.0.5 for Mac OS).

Estimation of the survival kinetics of E. coli O157:H7 using the 3-D model

Estimation of temperature change

The governing equations in the heating process were solved using the FEM in the COMSOL Multiphysics software (version 6.2). The simulation in the present study was for heat conduction in two-dimensional planar coordinates, as follows:

$${\displaystyle \rho C_p\frac{\mathrm{\partial }T}{\mathrm{\partial }t}=\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(kr\frac{\mathrm{\partial }T}{\mathrm{\partial }r}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(k\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right)\left[0\le r\le R,0\le z\le H\right]}$$ (5)

where T is the temperature (°C), t is time (s), r is the radial coordinate of the sample (m), z is the axial coordinate of the sample (m), R is the radius of the sample (0.045 m), H is half of the height of the sample (0.0085 m), ρ is the density of the sample (kg/m³), C_p is the specific heat of the sample [J/(kg·K)], and k is the thermal conductivity of the sample (W/m·K).

Because these thermal properties of the sample varied with the composition and temperature of the sample, they were calculated sequentially using the method described in “Determination of the thermophysical properties of the samples.” The computational load during the simulation was significantly reduced by setting up a region 1/4 of the actual shape for the calculation. The initial condition was a uniform temperature throughout the sample, as follows:

$$T\left(r,z,0\right)=T_i$$ (6)

where T_i is the initial temperature (°C). The boundary conditions were set as convective heat transfer between the water in the water bath and the sample at the top and sides [equation 7a and 7b ], and axisymmetry with r = 0 as the starting point [equation 8] and axisymmetry with z = 0 as the starting point [equation 9]

$${\displaystyle -\mathbf{n}\cdot \mathbf{q}=q_0}$$ (7a)

$${\displaystyle q_0=h\left(T_{\mathrm{e}\mathrm{x}\mathrm{t}}-T\right)}$$ (7b)

$${\displaystyle {\displaystyle \frac{\mathrm{\partial }T\left(0,z,t\right)}{\mathrm{\partial }r}}=0,[\mathrm{A}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}]}$$ (8)

$${\displaystyle {\displaystyle \frac{\mathrm{\partial }T\left(r,0,t\right)}{\mathrm{\partial }z}}=0,[\mathrm{A}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}]}$$ (9)

where n is the normal vector, q is the heating flux vector, q₀ is heat flux (W/m²), h is the heat transfer coefficient [W/(m²·K)], and T_ext is the temperature of the external fluid (°C). The values of T_i, T_ext, and h were 15°C, 62.5°C, and 2,000 W/(m²·K), respectively. The plastic bag in which the sample was packaged was assumed to be thin, and its thermal resistance was assumed to be very small and negligible. The accuracy of the temperature prediction was evaluated by comparing the changes in the center temperature calculated using equation 5 using the measured results.

Determination of the thermophysical properties of the samples

To solve equation 5 the thermal conductivity, specific heat, and density of the hamburger were calculated from the component compositions based on previous studies. For the mass fraction of each component, the values for imported beef thighs (lean meat), which were used as the material for ground beef, were used as shown in Table 2 (46). Literature values (47) were used to calculate the physical properties of each component at each temperature (Table 3). Using the mass fraction and density values of each component, the density was calculated using equation 10 as follows:

TABLE 2.

Mass fraction of each component in imported beef thigh meat (46)

	Composition (%)
	Moisture	Protein	Fat	Carbohydrate	Fiber	Ash
Mass fraction	74.2	21.2	4.3	0.4	0.0	1.0

TABLE 3.

Thermal property models for thermal conductivity, density, and specific heat of major components of foods (47)

Thermal property	Major component	Group models temperature function
Thermal conductivity	Water	kw = 5.7109 × 10−1 + 1.7625 × 10−3T−6.7036 × 10−6T 2
(W/m·K)	Protein	kp = 1.7881 × 10−1 + 1.1958 × 10−3T−2.7178 × 10−6T 2
	Fat	kf = 1.8071 × 10−1-2.7604 × 10−3T−1.7749 × 10−7T 2
	Carbohydrate	kc = 2.0141 × 10−1 + 1.3874 × 10−3T−4.3312 × 10−6T 2
	Ash	km = 3.2962 × 10−1 + 1.4011 × 10−3T−2.9069 × 10−6T 2
Density	Water	ρw= 9.9718 × 102 + 3.1439 × 10−3T−3.7574 × 10−3T 2
(kg/m3)	Protein	ρp = 1.3299 × 103–5.1840 × 10−1T
	Fat	ρf = 9.2559 × 102–4.1757 × 10−1T
	Carbohydrate	ρc = 1.5991 × 103–3.1046 × 10−1T
	Ash	ρm = 2.4238 × 10−1–2.8063 × 10−1T
Specific heat	Water	Cpw = 4.1762–9.0864 × 10−5T + 5.4731 × 10−6T 2
[J/(kg·K)]	Protein	Cpp = 2.0082 + 1.2089 × 10−3T−1.3129 × 10−6T 2
	Fat	Cpf = 1.9842 + 1.4733 × 10−3T−4.8008 × 10−6T 2
	Carbohydrate	Cpc = 1.5488 + 1.9625 × 10−3T−5.9399 × 10−6T 2
	Ash	Cpm = 1.0926 + 1.8896 × 10−3T−3.6817 × 10−6T 2

$${\displaystyle {\displaystyle \rho ={\displaystyle \frac{1}{\sum _{i=1}^n\left(x_i^w/{\rho }_i\right)}}}}$$ (10)

where i is each component, ρ_i is the density of each component (kg/m³), and $x_i^w$ is the mass fraction of each component. The thermal conductivity depends on the composition and spatial arrangement of the components. Several thermal conductivity estimation models have been proposed. According to Baghe-Khandan et al. (40), a series of models can be applied to estimate the thermal conductivity of ground beef. Kong et al. (48) also showed that the series model could be used to estimate the thermal conductivity of various meats, such as beef and pork. Therefore, a series model was adopted in this study, which was expressed as follows:

$${\displaystyle {\displaystyle {\displaystyle \frac{1}{k}}=\sum _{i=1}^n{\displaystyle \frac{x_i^v}{k_i}}}}$$ (11)

$${\displaystyle {\displaystyle x_i^v=\frac{x_i^w/{\rho }_i}{\sum _{j=1}^n\left(x_j^w/{\rho }_j\right)}}}$$ (12)

where i and j are each component, $x_i^v$ is the volume fraction of each component, and k_i is the thermal conductivity of each component (W/m·K), respectively. The specific heat of the sample at each temperature was calculated using equation 13 based on the additivity of the specific heat

$${\displaystyle {\displaystyle C_p=\sum _{i=1}^n{C}_{pi}{x}_i^w}}$$ (13)

where C_pi is the specific heat of each component [J/(kg·K)].

Estimation of the bacterial survival kinetics

Changes in the number of surviving E. coli O157:H7 cells in the entire sample were predicted from the temperature distribution estimated above and the changes during the heating process. Although it is difficult to experimentally determine the concentration of E. coli O157:H7 at a specific location in a sample, the average concentration of the entire sample after heat treatment can be determined. The average survival ratio in the whole hamburger was calculated by integrating the arithmetic survival numbers of E. coli O157:H7 in an infinitesimal region of the sample to obtain a kinetic prediction. The temperature change was calculated by solving equation 5, and both results were used to calculate the survival rate S(r, z, t) using equation (4). The average survival ratio for the whole sample was calculated using the COMSOL function. S (r, z, t) was calculated using the same procedure as in the 2-D model described in “Prediction of survival kinetics in non-isothermal conditions” using randomly sampled δ(T) and p.

SUPPLEMENTAL MATERIAL

The following material is available online at https://doi.org/10.1128/aem.00789-24.

Supplemental legends. aem.00789-24-s0001.docx.

Legends for Movies S1 and S2.

Movie S1. aem.00789-24-s0002.mp4.

Video of the predicted changes in the temperature gradient during heating.

Movie S2. aem.00789-24-s0003.mp4.

Video of the predicted changes in the survival ratio gradient during heating.

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