Staphylococcus aureus strains ATCC 12598, ATCC 25923, and ATCC 12600, obtained from the department of Food Science, University of Georgia, were used in this study. All strains were maintained at −70°C in tryptic soy broth (TSB, Difco, Sparks, MD, USA) containing 0.6% yeast extract (YE, Difco, USA) and 20% glycerol. For experimental purposes, the stock cultures of the strains were activated in tryptic soy broth (TSB; Difco, Detroit, MI, USA) at 35°C for 24 h in order to reach to the stationary phase. The bacteria were harvested by centrifugation (3,000 × g) at 4°C for 10 min, and then washed twice using 0.1% (w/v) sterile peptone water (PW; Difco, USA). The bacterial suspensions were mixed at equal concentrations to obtain a mixture with the final population level of approximately 5 log cfu/mL.

Gyungdan, a ball-shaped rice cake filled with sweet red bean paste and rolled in sesame seeds, were purchased from a local supermarket in Chuncheon, Korea. The pH and water activity of rice cake was approximately 6.14 and 0.975, respectively. The samples were cut in half using a sterile knife and placed on sterile aluminum foil in a lamella flow hood. The rice cakes were divided into 10 g samples with a balance to prepare for inoculation. For inoculation, 0.1 mL of the mixed culture (5 log cfu/mL) was applied to the samples by depositing droplets with a micropipettor. This procedure resulted in an initial pathogen inocula level of approximately 3 log cfu/g.

The inoculated samples were packaged with polyethylene and transferred into the growth chamber (BF-600GC, BioFree, Seoul, Korea). Uninoculated samples were used as a control. For the purpose of model development, the inoculated samples were stored at 5, 15, 25, 35, and 45°C at two different levels (50 and 80%) of relative humidity until they reached the stationary phase. As for model validation, the experiments were conducted at 10, 20, 30, and 40°C within the region of interpolation of the model. Sampling was generally carried out for enumeration based on the designed intervals, depending on the different incubation temperatures. Two samples were tested at each time interval. The growth studies were conducted in triplicate for each combination of conditions. Experimental period was set approximately 25 h at 35 and 45°C, 30 h at 30 and 40°C, and approximately 50 h at 20 and 25°C. However, longer periods were selected for the experiment at 5, 10, and 15°C.

At each sampling time, the 10 g rice cake samples to be tested were taken from the growth chamber and put into a sterile 400-mL stomaching bag with a filter (Nasco Whirl-Pak, Janesville, WI, USA) then mixed with 90 mL of 0.1% sterilized PW. Samples were then pummeled in a Seward Stomacher (400 Circulator, Seward, London, UK) at 200 rpm for 2 min. After homogenization, 1 mL of the sample suspension containing S. aureus was added into 9 mL 0.1% sterilized PW, and serial 10-fold dilutions were performed using 0.1% sterilized PW. Subsequently, 0.1 mL of the obtained samples or dilutions was plated in duplicate on Baird-Parker Agar (Difco Co., USA). After incubation of the plates at 35°C for 24 h, colonies were counted and the data were expressed as a logarithm of the population (log cfu/g). Means of cell populations from each treatment were calculated from three replications.

For each growth probability model development, each sampling point was scored with values of 0 and 1 to indicate whether or not to obtain 1-, 2-, 3-, and 4-log increases of S. aureus growth, respectively. The probability of a 1-, 2-, 3-, and 4-log increases of S. aureus from the growth data on rice cakes was collected and fitted to a logistic regression model using R software (A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/). The analyses were made with the glm() function that fits generalized linear models in R-software package STATS. The initial model for fitting had the following form after a minor modification according to the approach described by Ratkowsky and Ross (1995), Koseki et al. (2009), and Agresti (2007): (5)Logit(P)=α0+α1Temp+α2 ·ln(Time) where P is the probability of an arbitrary log-increase based on S. aureus growth, Logit(P)=ln(P1-P), α₀−α₂ are the coefficients to be estimated, Temp is the storage temperature, and Time is the time at which the increase of S. aureus growth reached the set values.

To evaluate the goodness-of-fit the developed model, the maximum rescaled R-square statistic, the Hosmer-Lemeshow goodness-of-fit statistic, and the receiver operating characteristic (ROC) curve were used (Agresti, 2007). The maximum rescaled R² for use with binomial error was proposed as a generalization of the coefficient of determination R that is commonly used in regression applications involving normally distributed error (Nagelkerke, 1991; Tienungoon et al., 2000). The Hosmer-Lemeshow goodness-of-fit statistic, which involves grouping objects into a contingency table and calculating a Pearson chi-square statistic, was proposed as a means of estimating goodness of fit (Tienungoon et al., 2000). Small values of the statistic (large P-values) indicate a good fit of the model to the data. The area under the ROC curve, c, is a measure of discrimination, obtained from a plot of sensitivity, i.e., the proportion of observed events that were correctly predicted to be events, against the complement of specificity, i.e., the proportion of nonevents that were correctly predicted to be nonevents. The closer the value of c is to 1, the greater is the discrimination. In epidemiological studies, a c value of >0.7 is considered acceptable discrimination, a c value of >0.8 as good discrimination, and a c value of >0.9 as excellent discrimination (Lemeshow and Jean-Roger, 1994).

The performance and reliability of the developed models should be validated before actual application. The validation step was conducted using experimental data which was not used for model development. In the validation step, several statistical indices, such as bias factor (B_f, Equation 6), accuracy factor (A_f, Equation 7), the root mean square error (RMSE, Equation 8; McKellar and Lu, 2004) and %standard error of prediction (%SEP, Equation 9; Garcia-Gimeno et al., 2005) were employed as follows: (6)Bf=10(∑i=1nlog(μobserved/μpredicted)/n) (7)Af=10(∑i=1n|log(μpredicted/μobserved)|/n) (8)RMSE=∑(μobserved−μpredicted)2n (9)%SEP=100Average(μobserved)∑(μobserved−μpredicted)2n where n is the number of observations, μ_observed is the observed value, and μ_predicted is the predicted value.

Differential secondary model performance was evaluated using the RMSE. In addition, the acceptable prediction zone method was used for evaluation of the model performance (Oscar, 2005). The prediction errors or relative errors (REs) for the individual fitted cases were calculated according to the following equation: (10)RE=Predicted − ObservedPredicted

The model gives fail-safe predictions when the RE is <0, while fail-dangerous predictions at RE values >0. The proportion of RE (pRE) in the acceptable prediction zone of RE −0.6 to 0.3, wherein the proportion of ≥0.70 indicated an acceptable model, was used to evaluate the performance of the obtained model.

All the data from the three replicates were expressed as log colony forming units per gram (log CFU/g) and used together for model development. Statistical analysis was performed using IBM SPSS statistics 20 (IBM Corporation, New York, USA). Data were expressed as means ± standard deviation (SD). All data were analyzed using Analysis of variance (ANOVA). Statistical significance of growth parameters was analyzed by t-test at a significance level of 0.05. Values of validation indices were calculated in a Microsoft Excel spreadsheet.

The experimental data of S. aureus on rice at all the designed temperatures for model development and validation were collected and presented in Figures 1, 2, respectively. Experimental observations revealed S. aureus to be undetectable on the uninoculated rice cake samples (control). S. aureus did not grow well on the rice cake stored at 5 and 10°C (Figure 3), and a decline in the S. aureus cells was observed at 5°C. This was similar with previous studies which indicated S. aureus could grow at temperatures as low as 7°C (ICMSF, 1980). In contrast, no growth was observed at 10°C for the storage period of up to 7 days. The reason may be because the target samples were different food matrices, and Staphylococci are particularly sensitive to nutrient depletion. The data at these lower incubation temperatures (5 and 10°C) were not included in the development and validation of the model. The final cell populations of S. aureus at 15, 25, 35, and 45°C were 7.45, 7.55, 7.44, and 7.65 log CFU/mL, respectively, indicating that higher temperatures led to higher final cell populations. Among them, the higher values at 15 and 25°C are due to the higher initial population levels. Similar results were demonstrated in our previous research, in which the temperature was more influential on pathogen growth on the food matrix than humidity (Wang and Oh, 2012; Wang et al., 2012).

To simulate growth of S. aureus on rice cake under real temperature profile, we calculated the number of S. aureus by using simultaneous differential equations (Equations 2, 3). The physiological state parameter q₀ was determined by average of the estimated fitted parameters of four iso-thermal growth curves. The q₀ of the tested growth curves of S. aureus was determined as 0.20 ± 0.14. Figure 4 demonstrated one of the representing results under fluctuating temperature condition. The RMSE for S. aureus growth on rice cake was 0.218, while the acceptable prediction rate was 90.9%. These indices exhibited acceptable performance of the developed model for S. aureus growth on rice cake under the tested temperatures. In the present study, just one of the representing fluctuating temperature condition was monitored. The model was not tested for significant changes in temperature. Although the predictive capabilities of the model under dynamic temperature were not conducted, the calculation procedure would enable to correspond to real temperature history during distribution.

Fujikawa et al. (2004) developed a new logistic model for bacterial growth at dynamic temperatures using a numerical solution with the fourth-order Runge–Kutta method and demonstrated that the newly developed model could successfully predict Escherichia coli and Salmonellae growth curves for various patterns of the temperature history. Subsequently, S. aureus growth in sterilized milk were successfully predicted at various patterns of varying temperature using the newly developed logistic model, while its enterotoxin amounts predicted with the developed regression line for Staphylococcal enterotoxins were higher than the observed values (Fujikawa and Morozumi, 2006). In addition, comparison of the dynamic models based on the Baranyi model and modified logistic model could produce almost the same growth curves for varying temperature histories (Fujikawa and Morozumi, 2005).

In order to determine the effect of temperature on the growth of S. aureus on rice cake during the storage period, the probability of the time required to reach a certain level was determined using the logistic regression model described in Equation (4). The estimated parameters of the logistic regression models for 1-, 2- 3-, and 4-log increases of S. aureus expansion on rice cake are presented in Table 3. The information of goodness-of-fit of the model, such as 95% confidence intervals of the parameters, maximum rescaled R², area under ROC curve, and Hosmer-Lemeshow goodness of fit are also listed in Table 3. According to the performance statistics, the derived parameters for the developed logistic regression models showed excellent discrimination for all the log increases.

The cumulative probability distributions predicted by the developed models listed in Table 3 are shown in Figure 5 for the 1-, 2-, 3-, and 4-log increases of S. aureus on rice cake, respectively. From the figure, the probability of the time required to reach a certain level of S. aureus contamination on the rice cakes at different temperatures could be determined. Meanwhile, the probability of the time required to reach different levels of S. aureus on rice cake at a certain temperature is presented in Figure 6.

Using the obtained model, the Logit (P) values can be obtained, and the probabilities of S. aureus growth on rice cake can be estimated. Referred the reported limits on the growth of bacteria (Presser et al., 1998; Tienungoon et al., 2000; Le Marc et al., 2005; Hwang and Juneja, 2011) that p ≤ 0.1 indicates an “unlikely to grow” or “no-growth” region, p > 0.5 indicates a “likely to grow” or “growth” region, and p values between 0.1 and 0.5 indicate an “uncertainty” region, in the present study, the criterion of a certain log increases of S. aureus was set to be P = 0.5 and P = 0.1. That is to say, as for a certain level of logistic regression model, P ≤ 0.1 indicates an “unlikely reach” region of a certain log increase, P > 0.5 indicates a “reach” region, and P-values between 0.1 and 0.5 indicate a “likely reach” region (Figure 5). According to this criterion, we could derive an equation through transformation from the obtained logistic regression models shown in Table 3 to calculate the time to reach a certain log increase of S. aureus at different temperatures within the limits of the experimental design. The equations to calculate how long time to reach 1-, 2-, 3-, and 4-log increase of S. aureus were listed as following: (12)t=exp ((22.317−0.355∗Temp)/5.835) (13)t=exp ((32.48−0.441∗Temp)/7.667) (14)t=exp ((33.43−0.435∗Temp)/7.077) (15)t=exp ((29.50−0.400∗Temp)/4.998) where Temp is the storage temperature, and t is the time at which 1-, 2-, 3-, and 4-log increases of S. aureus reached, respectively.

Time and temperature abuse would lead to results of pathogenic bacteria growth and toxin formation. S. aureus concentrations of more than 10⁵ CFU/mL were unacceptable, since numbers of S. aureus higher than 10⁵ most often lead to human illness (Food and Drug Administration, 2012). In addition, it is extremely likely to produce Staphylococcal enterotoxins under specific environmental conditions when the S. aureus level reaches 10⁵ CFU/mL (Heidinger et al., 2009). Fujikawa and Morozumi (2006) reported that temperature dependency of the rate constant of staphylococcal enterotoxin A (ng/mL/h) could be calculated as follows: (16)ptoxin=0.0376∗T−0.559 where p_toxin is the production rate of staphylococcal enterotoxin A, T is temperature. Based on Equation (16), enterotoxin production rate at each temperature would be estimated. According to Equation (14), it is easy to know the time to reach 3-log increase of S. aureus growth at different temperatures within the limits of the experimental design. As shown in Figure 5c, the area above the solid line was “reach” region, and the time to reach 3-log increase of S. aureus growth at 15, 25, 35, and 45°C were 44.8, 24.2, 13.1, and 7.1 h, respectively. At the same time, the probability to reach 3-log increase could be obtained. Finally, the hazardous levels at each temperature could be evaluated considering the probability models and the enterotoxin production rate. If we could check the initial contamination level of rice cake, it is convenient to monitor the status of rice cake products. These logistic regression models could identify the shelf life of rice cake based on the initial levels of S. aureus contamination. From a food industry standpoint, to control the growth conditions of pathogens contributing to levels exceeding 10⁵ will be of key importance. Therefore, the developed probability models will be very useful for food safety management and microbiological risk assessment of S. aureus on rice cake in Korea.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.