Development and validation of a dynamic growth model for Listeria monocytogenes in fluid whole milk

Listeria monocytogenes, a psychrotrophic microorganism, has been the cause of several food-borne illness outbreaks, including those traced back to pasteurized fluid milk and milk products. This microorganism is especially important because it can grow at storage temperatures recommended for milk (< or =7 degrees C). Growth of L. monocytogenes in fluid milk depends to a large extent on the varying temperatures it is exposed to in the postpasteurization phase, i.e., during in-plant storage, transportation, and storage at retail stores. Growth data for L. monocytogenes in sterilized whole milk were collected at 4, 6, 8, 10, 15, 20, 25, 30, and 35 degrees C. Specific growth rate and maximum population density were calculated at each temperature using these data. The data for growth rates versus temperature were fitted to the Zwietering square root model. This equation was used to develop a dynamic growth model (i.e., the Baranyi dynamic growth model or BDGM) for L. monocytogenes based on a system of equations which had an intrinsic parameter for simulating the lag phase. Results from validation of the BDGM for a rapidly fluctuating temperature profile showed that although the exponential growth phase of the culture under dynamic temperature conditions was modeled accurately, the lag phase duration was overestimated. For an alpha0 (initial physiological state parameter) value of 0.137, which corresponded to the mean temperature of 15 degrees C, the population densities were underpredicted, although the experimental data fell within the narrow band calculated for extreme values of alpha0. The maximum relative error between the experimental data and the curve based on an average alpha0 value was 10.42%, and the root mean square error was 0.28 log CFU/ml.     

Modeling Staphylococcus aureus growth and enterotoxin production in milk

Staphylococcus aureus growth and its enterotoxin production in sterilized milk were modeled with a modification of a new logistic model recently developed by us. The modified model and the Baranyi model described the early exponential phase of a growth curve more accurately than the previous model, at constant temperatures from 14 to 36.5 degrees C. The amount of toxin in milk increased linearly with time from the time the cell population reached about 10(6.5) cfu/ml. The rate of toxin production linearly increased at temperatures between 14 and 32 degrees C. From parameter values obtained at the constant temperatures, the model successfully predicted bacterial growth in the milk at a varying temperature. For toxin level estimation, we postulated that the rate of toxin production might be regulated with the temperature after the cell concentration reached 10(6.5) cfu/ml; the time point when the cell concentration reached that value was predicted with the modified growth model. Introduction of a correction factor in the toxin estimation successfully predicted the toxin level in milk at a varying temperature. These results showed that this prediction system consisting of the modified model and the toxin production algorithm might be a useful tool for modeling bacterial growth and its metabolite production in liquid foods.     

Partitioning of the variance in the growth parameters of Erwinia carotovora on vegetable products

The objective of this paper was to estimate and partition the variability in the microbial growth model parameters describing the growth of Erwinia carotovora on pasteurised and non-pasteurised vegetable juice from laboratory experiments performed under different temperature-varying conditions. We partitioned the model parameter variance and covariance components into effects due to temperature profile and replicate using a maximum likelihood technique. Temperature profile and replicate were treated as random effects and the food substrate was treated as a fixed effect. The replicate variance component was small indicating a high level of control in this experiment. Our analysis of the combined E. carotovora growth data sets used the Baranyi primary microbial growth model along with the Ratkowsky secondary growth model. The variability in the microbial growth parameters estimated from these microbial growth experiments is essential for predicting the mean and variance through time of the E. carotovora population size in a product supply chain and is the basis for microbiological risk assessment and food product shelf-life estimation. The variance partitioning made here also assists in the management of optimal product distribution networks by identifying elements of the supply chain contributing most to product variability.     

Mathematical modelling for predicting the growth of Pseudomonas spp. in poultry under variable temperature conditions

A dynamic growth model under variable temperature conditions was implemented and calibrated using raw data for microbial growth of Pseudomonas spp. in poultry under aerobic conditions. The primary model was implemented using measurement data under a set of fixed temperatures. The two primary models used for predicting the growth under constant temperature conditions were: Baranyi and modified Gompertz. For the Baranyi model the maximum specific growth rate and the lag phase at constant environmental conditions are expressed in exact form and it has been shown that in limit case when maximal cells concentration is much higher than the initial concentration the maximum specific growth rate is approximately equal to the specific growth rate. The model parameters are determined in a temperature range of 2-20 degrees C. As a secondary model the square root model was used for maximum specific growth rate in both models. In both models the main assumption, that the initial physiological state of the inoculum is constant and independent of the environmental parameters, is used, and a free parameter was implemented which was determined by minimizing the mean square error (MSE) relative to the measurement data. Two temperature profiles were used for calibration of the models on the initial conditions of the cells.     

Prediction of pathogen growth on iceberg lettuce under real temperature history during distribution from farm to table

The growth of pathogenic bacteria Escherichia coli O157:H7, Salmonella spp., and Listeria monocytogenes on iceberg lettuce under constant and fluctuating temperatures was modelled in order to estimate the microbial safety of this vegetable during distribution from the farm to the table. Firstly, we examined pathogen growth on lettuce at constant temperatures, ranging from 5 to 25 degrees C, and then we obtained the growth kinetic parameters (lag time, maximum growth rate (micro(max)), and maximum population density (MPD)) using the Baranyi primary growth model. The parameters were similar to those predicted by the pathogen modelling program (PMP), with the exception of MPD. The MPD of each pathogen on lettuce was 2-4 log(10) CFU/g lower than that predicted by PMP. Furthermore, the MPD of pathogens decreased with decreasing temperature. The relationship between mu(max) and temperature was linear in accordance with Ratkowsky secondary model as was the relationship between the MPD and temperature. Predictions of pathogen growth under fluctuating temperature used the Baranyi primary microbial growth model along with the Ratkowsky secondary model and MPD equation. The fluctuating temperature profile used in this study was the real temperature history measured during distribution from the field at harvesting to the retail store. Overall predictions for each pathogen agreed well with observed viable counts in most cases. The bias and root mean square error (RMSE) of the prediction were small. The prediction in which mu(max) was based on PMP showed a trend of overestimation relative to prediction based on lettuce. However, the prediction concerning E. coli O157:H7 and Salmonella spp. on lettuce greatly overestimated growth in the case of a temperature history starting relatively high, such as 25 degrees C for 5 h. In contrast, the overall prediction of L. monocytogenes under the same circumstances agreed with the observed data.     

Model on the microbial quality change of seasoned soybean sprouts for on-line shelf life prediction

The growth of aerobic bacteria on Korean seasoned soybean sprouts was modelled as a function of temperature to estimate microbial spoilage and shelf life on a real-time basis under dynamic storage conditions. Counts of aerobic bacteria on seasoned soybean sprouts stored at constant temperatures between 0 degrees C and 15 degrees C were recorded. The bootstrapping method was applied to generate many resampled data sets of mean microbial plate counts that were then used to estimate the parameters of the microbial growth model of Baranyi and Roberts. The distributions of the model parameters were quantified, and their temperature dependencies were expressed as mathematical functions. When the temperature functions of the parameters were incorporated into differential equations describing microbial growth, predictions of microbial growth under fluctuating temperature conditions were similar to observed microbial growth.     

不同温度下椰汁中金黄色葡萄球菌的生长动力学模型比较

探讨不同温度下椰汁中金黄色葡萄球菌的生长预测模型。将菌悬液接种到椰汁中,测定不同温度(20、25、30、36℃)下的生长数据。使用Matlab软件拟合得到修正Gompertz(MGompertz)、修正Logistic(MLloistic)和Baranyi模型,比较残差和拟合度选择最优一级模型,并拟合出生长参数。用平方根和二次多项式方程建立二级模型,通过相关系数、偏差因子和准确因子对二级模型进行检验。在20~36℃下,Baranyi模型拟合出的各个拟合度最优,Baranyi模型适宜作为模拟金黄色葡萄球菌在椰汁中生长的一级预测模型。二次多项式相较于平方根模型可以更好地表达温度与最大比生长速率及延滞期的关系。因此选择Baranyi模型和二次多项式模型描述不同温度下椰汁中金黄色葡萄球菌的生长。     

Growth curve prediction from optical density data

A fundamental aspect of predictive microbiology is the shape of the microbial growth curve and many models are used to fit microbial count data, the modified Gompertz and Baranyi equation being two of the most widely used. Rapid, automated methods such as turbidimetry have been widely used to obtain growth parameters, but do not directly give the microbial growth curve. Optical density (OD) data can be used to obtain the specific growth rate and if used in conjunction with the known initial inocula, the maximum population data and knowledge of the microbial number at a predefined OD at a known time then all the information required for the reconstruction of a standard growth curve can be obtained. Using multiple initial inocula the times to detection (TTD) at a given standard OD were obtained from which the specific growth rate was calculated. The modified logistic, modified Gompertz, 3-phase linear, Baranyi and the classical logistic model (with or without lag) were fitted to the TTD data. In all cases the modified logistic and modified Gompertz failed to reproduce the observed linear plots of the log initial inocula against TTD using the known parameters (initial inoculum, MPD and growth rate). The 3 phase linear model (3PLM), Baranyi and classical logistic models fitted the observed data and were able to reproduce elements of the OD incubation-time curves. Using a calibration curve relating OD and microbial numbers, the Baranyi equation was able to reproduce OD data obtained for Listeria monocytogenes at 37 and 30°C as well as data on the effect of pH (range 7.05 to 3.46) at 30°C. The Baranyi model was found to be the most capable primary model of those examined (in the absence of lag it defaults to the classic logistic model). The results suggested that the modified logistic and the modified Gompertz models should not be used as Primary models for TTD data as they cannot reproduce the observed data.     

LABORATORY EVALUATION OF PAPERBOARD CARTONS FOR THE NONREFRIGERATED STORAGE OF STERILIZED MILK1,2

The suitability of various carton materials for the nonrefrigerated storage of sterilized milk was investigated. One quart paperboard cartons were fabricated from the same base sheet of stock but varied in the type of sizing used to make them resistant to penetration by liquids and whether or not they were aluminum foil-lined. They were preformed and sterilized with ethylene oxide. The four types of paperboard were: (a) rosin (sizing) paperboard (R); (b) rosin paperboard with foil lining (RF); (c) cyanasize (sizing) juice paperboard (CJ); and (d) cyanasizejuice paperboard with foil lining (CJF). Each carton was aseptically filled and sealed, in a glovebox. Incubation was carried out at 20°C for up to nine weeks. Every week five cartons of each type were randomly selected and the milk tested for microbial stability and flavor. The candidate cartons were also tested for degradation of the physical characteristics of static bulge, wicking, tensile strength, and stiffness. Of these, it appears as if selection of carton type will be determined mostly by wicking resistance. The most acceptable carton type is CJF, which had minimal wicking, acceptable bulge, acceptable stiffness, and acceptable tensile strength during the testing period.     

Statistical evaluation of mathematical models for microbial growth

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO(2) (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.     

Comparison of the thermal inactivation of Bacillus subtilis spores in foods using the modified Weibull and Bigelow equations

The association of a modified Weibull model and Bigelow model was applied to the thermal inactivation of Bacillus subtilis spores heated in phosphate buffer, milk, kayu (a Japanese style rice porridge) and soy sauce as well. The inactivation kinetics presented a light downward concave profile, the acidic pH increased the efficiency of the heat treatment but on the opposite, lesser the water activity, weaker was the efficiency. The heat treatment kinetics observed in milk, soy sauce and kayu were greatly different from each other, while no large difference between sterilized whole milk, UHT whole milk, sterilized skim milk and UHT skim milk, were observed. The model established in buffer system allowed heat treatment in milk products to be simulated although it could not be employed to describe the inactivation of B. subtilis spores in soy sauce and kayu. For these two latter products, the food itself had to be introduced in the model as a parameter. Finally, this approach combining primary model (to simulate inactivation kinetics) and secondary model (to introduce temperature, pH, a_w and food matrix effect) seemed available for food application, nevertheless validations of results such as challenge-tests, must be performed before it is put to routine use.     

Mathematical Modeling of Lactobacillus Viridescens Growth in Vacuum Packed Sliced Ham under non Isothermal Conditions

Lactic acid bacteria (LAB) are responsible for the spoilage of vacuum packed meat products, as ham. Temperature is the main factor affecting the microbial dynamics and its variation during the production, distribution and storage of foods is considerable. Thus, the use of mathematical models to describe the microbial behavior under variable temperatures can be very useful in predicting the food shelf life. This study evaluated the growth of Lactobacillus viridescens in sliced ham under non isothermal conditions, and assessed the predictive ability of the Baranyi and Roberts model using parameters obtained isothermally in culture medium (MRS). To obtain the BAL growth, the fresh ham piece was sterilized, sliced, inoculated with bacteria and stored in a temperature-controlled incubator. For the establishment of the secondary models, the primary model parameters were obtained isothermally in the culture medium at 4, 8, 12, 16, 20 and 30° C, in which there was no lag phase observed; the square root model was selected to describe the dependence of the μ_max parameter (maximum specific growth rate) with the temperature, and the y_max parameter (maximum population) was represented by an average because there was no significant influence of the temperature. The mathematical models were validated with L. viridescens growth data in ham under five variable temperature conditions (NI-1 (4-8-12-16°C), NI-2 (12-16-20-25°C), NI-3 (25-20-16-12-8-4°C), NI-4 (16-12-8-4°C) and NI-5 (12-8-4-8-12°C)), and its predictive ability were assessed through statistical indexes (bias factor, accuracy factor and RMSE), with good results (bias factor between 0.9450 and 1.0326; accuracy factor between 1.0382 and 1.0682, and RMSE between 0.7641 and 1.3317), especially in increasing temperature, where the prediction was safe. The validated model can be used to estimate the shelf life of a commercial ham under different temperature conditions.     

生孢梭菌孢子生长与失活模型的统一化研究

为探讨预测微生物生长和失活的预测模型统一化问题,研究将生孢梭菌孢子热失活“镜像化”曲线用描述微生物生长的Gompertz 模型和Baranyi 模型进行模拟,并通过标准预测误对两种模型进行比较。实验表明：两种模型都能较好的模拟生孢梭菌孢子热失活,但t 检验分析差异不显著,标准预测误比较表明Gompertz 模型优于Baranyi 模型。建议用Gompertz 模型统一描述生孢梭菌孢子生长和失活情况。     

Biological implications from an autonomous version of Baranyi and Roberts growth model

An autonomous version of Baranyi and Roberts model (Int. J. Food Microbiology, 23, 1994, pp. 277-294) is developed and analyzed to reveal some subtle points, which are difficult to observe accurately from its equivalent non-autonomous form. In particular we are able to provide a meaningful interpretation to the "physiological state of the cells at inoculation", a parameter introduced by Baranyi and Roberts [Baranyi, J., Roberts, T. A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277-294] that has a profound impact on microbial growth but is not a direct measurable quantity. In addition, the analysis shows that the transient growth depends on the initial cell concentration and the initial growth rate, but is independent of "the history of the cells" and depends only indirectly (via the initial growth rate) on the previous (pre-inoculation) environment. The stationary solution is independent of the initial conditions. A new, more natural, and biologically meaningful formulation of LAG duration is being suggested in terms of initial conditions being in the neighborhood of one of the unstable stationary points revealed by this autonomous version of the model.     

Dynamic modeling of Listeria monocytogenes growth in pasteurized vanilla cream after postprocessing contamination

A product-specific model was developed and validated under dynamic temperature conditions for predicting the growth of Listeria monocytogenes in pasteurized vanilla cream, a traditional milk-based product. Model performance was also compared with Growth Predictor and Sym'Previus predictive microbiology software packages. Commercially prepared vanilla cream samples were artificially inoculated with a five-strain cocktail of L. monocytogenes, with an initial concentration of 102 CFU g(-1), and stored at 3, 5, 10, and 15 degrees C for 36 days. The growth kinetic parameters at each temperature were determined by the primary model of Baranyi and Roberts. The maximum specific growth rate (mu(max)) was further modeled as a function of temperature by means of a square root-type model. The performance of the model in predicting the growth of the pathogen under dynamic temperature conditions was based on two different temperature scenarios with periodic changes from 4 to 15 degrees C. Growth prediction for dynamic temperature profiles was based on the square root model and the differential equations of the Baranyi and Roberts model, which were numerically integrated with respect to time. Model performance was based on the bias factor (B(f)), the accuracy factor (A(f)), the goodness-of-fit index (GoF), and the percent relative errors between observed and predicted growth. The product-specific model developed in the present study accurately predicted the growth of L. monocytogenes under dynamic temperature conditions. The average values for the performance indices were 1.038, 1.068, and 0.397 for B(f), A(f), and GoF, respectively for both temperature scenarios assayed. Predictions from Growth Predictor and Sym'Previus overestimated pathogen growth. The average values of B(f), A(f), and GoF were 1.173, 1.174, and 1.162, and 1.267, 1.281, and 1.756 from Growth Predictor and Sym'Previus, respectively.     

基于Baranyi模型的波动温度下鲐鱼微生物生长动力学模型

以鲐鱼为研究对象,进行恒温下细菌总数的计数并建立基于Baranyi模型的微生物生长动力学模型。对Baranyi模型进行改进,构建波动温度下的微生物生长动力学模型,并运用模拟实验对模型进行验证,验证结果是偏差因子为1.23,准确因子为1.12,表明所构建的微生物生长动力学模型能够很好的拟合波动温度下鲐鱼中微生物的变化情况。     

Introducing optimal experimental design in predictive modeling: a motivating example

Predictive microbiology emerges more and more as a rational quantitative framework for predicting and understanding microbial evolution in food products. During the mathematical modeling of microbial growth and/or inactivation, great, but not always efficient, effort is spent on the determination of the model parameters from experimental data. In order to optimize experimental conditions with respect to parameter estimation, experimental design has been extensively studied since the 1980s in the field of bioreactor engineering. The so-called methodology of optimal experimental design established in this research area enabled the reliable estimation of model parameters from data collected in well-designed fed-batch reactor experiments. In this paper, we introduce the optimal experimental design methodology for parameter estimation in the field of predictive microbiology. This study points out that optimal design of dynamic input signals is necessary to maximize the information content contained within the resulting experimental data. It is shown that from few dynamic experiments, more pertinent information can be extracted than from the classical static experiments. By introducing optimal experimental design into the field of predictive microbiology, a new promising frame for maximization of the information content of experimental data with respect to parameter estimation is provided. As a case study, the design of an optimal temperature profile for estimation of the parameters D(ref) and z of an Arrhenius-type model for the maximum inactivation rate kmax as a function of the temperature, T, was considered. Microbial inactivation by heating is described using the model of Geeraerd et al. (1999). The need for dynamic temperature profiles in experiments aimed at the simultaneous estimation of the model parameters from measurements of the microbial population density is clearly illustrated by analytical elaboration of the mathematical expressions involved on the one hand, and by numerical simulations on the other.     

Heat transfer models for predicting Salmonella enteritidis in shell eggs through supply chain distribution

ABSTRACT:  Egg and egg preparations are important vehicles for Salmonella enteritidis infections. The influence of time–temperature becomes important when the presence of this organism is found in commercial shell eggs. A computer-aided mathematical model was validated to estimate surface and interior temperature of shell eggs under variable ambient and refrigerated storage temperature. A risk assessment of S. enteritidis based on the use of this model, coupled with S. enteritidis kinetics, has already been reported in a companion paper published earlier in JFS . The model considered the actual geometry and composition of shell eggs and was solved by numerical techniques (finite differences and finite elements). Parameters of interest such as local ( h ) and global ( U ) heat transfer coefficient, thermal conductivity, and apparent volumetric specific heat were estimated by an inverse procedure from experimental temperature measurement. In order to assess the error in predicting microbial population growth, theoretical and experimental temperatures were applied to a S. enteritidis growth model taken from the literature. Errors between values of microbial population growth calculated from model predicted compared with experimentally measured temperatures were satisfactorily low: 1.1% and 0.8% for the finite difference and finite element model, respectively.     

Model comparison for Escherichia coli growth in pouched food

We recently studied the growth characteristics of Escherichia coli cells in pouched mashed potatoes (Fujikawa et al., J. Food Hyg. Soc. Japan, 47, 95-98 (2006)). Using those experimental data, in the present study, we compared a logistic model newly developed by us with the modified Gompertz and the Baranyi models, which are used as growth models worldwide. Bacterial growth curves at constant temperatures in the range of 12 to 34 degrees C were successfully described with the new logistic model, as well as with the other models. The Baranyi gave the least error in cell number and our model gave the least error in the rate constant and the lag period. For dynamic temperature, our model successfully predicted the bacterial growth, whereas the Baranyi model considerably overestimated it. Also, there was a discrepancy between the growth curves described with the differential equations of the Baranyi model and those obtained with DMfit, a software program for Baranyi model fitting. These results indicate that the new logistic model can be used to predict bacterial growth in pouched food.