Modeling the effect of temperature on growth of Salmonella in chicken

Growth data of Salmonella in chicken were collected at several isothermal conditions (10, 15, 20, 25, 28, 32, 35, 37, 42, and 45 degrees C) and were then fitted into primary models, namely the logistic model, modified Gompertz model and Baranyi model. Measures of goodness-of-fit such as mean square error, pseudo-R(2), -2 log likelihood, Akaike's information, and Sawa's Bayesian information criteria were used for comparison for these primary models. Based on these criteria, modified Gompertz model described growth data the best, followed by the Baranyi model, and then the logistic model. The maximum growth rates obtained from each primary model were then modeled as a function of temperature using the modified Ratkowsky model. Pseudo-R(2) values for this secondary model describing growth rate obtained from Baranyi, modified Gompertz, and logistic models were 0.999, 0.980, and 0.990, respectively. Mean square error values for corresponding models were 0.0002, 0.0008, and 0.0009, respectively. Both measures clearly show that the Baranyi model performed better than the modified Gompertz model or the logistic model.     

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.     

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.     

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。     

When is simple good enough: a comparison of the Gompertz,Baranyi, and three-phase linear models for fitting bacterial growth curves

The use of primary mathematical models with curve fitting software is dramatically changing quantitative food microbiology. The two most widely used primary growth models are the Baranyi and Gompertz models. A three-phase linear model was developed to determine how well growth curves could be described using a simpler model. The model divides bacterial growth curves into three phases: the lag and stationary phases where the specific growth rate is zero (gm=0), and the exponential phase where the logarithm of the bacterial population increases linearly with time (gm=constant). The model has four parameters: N_o(Log₈of initial population density), NMAX(Log₈of final population density), tLAG(time when lag phase ends), and tMAX(time when exponential phase ends). A comparison of the linear model was made against the Baranyi and Gompertz models, using established growth data forEscherichia coli0157:H7. The growth curves predicted by the three models showed good agreement. The linear model was more ‘robust' than the others, especially when experimental data were minimal. The physiological assumptions underlying the linear model are discussed, with particular emphasis on assuring that the model is consistent with bacterial behavior both as individual cells and as populations. It is proposed that the transitional behavior of bacteria at the end of the lag phase can be explained on the basis of biological 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.     

低温条件下冷却猪肉中假单胞菌生长模型的比较分析

为了确定拟合冷却猪肉中假单胞菌低温下生长的最适模型,分别对低温(0、5、10℃)条件下托盘和真空包装冷却猪肉中假单胞菌的生长特点进行分析,应用修正的Gompertz、Baranyi及Huang模型对其进行拟合,通过残差和拟合度(RSS、AIC、RSE)等统计指标比较3种模型的拟合能力,分析不同模型拟合假单胞菌生长的差别。结果表明：低温托盘和真空包装条件下假单胞菌在延滞期出现了明显的菌数下降现象,随后呈现“S”形生长;0℃条件下Baranyi模型拟合出最小的RSS、AIC、RSE值,分别是5.2933、－54.0428、0.1708;而修正的Gompertz模型和Huang模型分别在5℃和10℃条件下拟合出最小的RSS、AIC、RSE值,分别是17.7372、－18.9098、0.5068和13.0410、－22.4848、0.4207。拟合冷却猪肉中假单胞菌生长的最适模型0℃是Baranyi模型,5℃是修正的Gompertz模型,10℃是Huang模型。因此,在冷却猪肉腐败菌预测时,不同温度条件下应该选择最适合的模型而不是单一的模型来预测假单胞菌的生长。     

Growth kinetics of Listeria monocytogenes in broth and beef frankfurters--determination of lag phase duration and exponential growth rate under isothermal conditions

ABSTRACT:  The objective of this study was to develop a new kinetic model to describe the isothermal growth of microorganisms. The new model was tested with Listeria monocytogenes in tryptic soy broth and frankfurters, and compared with 2 commonly used models—Baranyi and modified Gompertz models. Bias factor (BF), accuracy factor (AF), and root mean square errors (RMSE) were used to evaluate the 3 models. Either in broth or in frankfurter samples, there were no significant differences in BF (approximately 1.0) and AF (1.02 to 1.04) among the 3 models. In broth, the mean RMSE of the new model was very close to that of the Baranyi model, but significantly lower than that of the modified Gompertz model. However, in frankfurters, there were no significant differences in the mean RMSE values among the 3 models. These results suggest that these models are equally capable of describing isothermal bacterial growth curves. Almost identical to the Baranyi model in the exponential and stationary phases, the new model has a more identifiable lag phase and also suggests that the bacteria population would increase exponentially until the population approaches to within 1 to 2 logs from the stationary phase. In general, there is no significant difference in the means of the lag phase duration and specific growth rate between the new and Baranyi models, but both are significantly lower than those determined from the modified Gompertz models. The model developed in this study is directly derived from the isothermal growth characteristics and is more accurate in describing the kinetics of bacterial growth in foods.     

Comparison of the Baranyi model with the modified Gompertz equation for modelling thermal inactivation of Listeria monocytogenes Scott A

The Baranyi model was used to fit the four commonly observed survival curves: linear curves, those with a lag phase, those with a tailing phase and sigmoidal curves. It was validated by using published experimental data for thermal inactivation of Listeria monocytogenes Scott A heated in infant formula and compared with the modified Gompertz equation. For the prediction performance, the Baranyi model was better and more robust than the modified Gompertz equation.     

Determination of the growth limits and kinetic behavior of Listeria monocytogenes in a sliced cooked cured meat product: validation of the predictive growth model under constant and dynamic temperature storage conditions

To describe the growth limits of Listeria monocytogenes NCTC10527 in a sliced vacuum-packaged cooked cured meat product, the binary logistic regression model was used to develop an equation to determine the probability of growth or no growth of L. monocytogenes as a function of temperature (from 0 to 10 degrees C) and water activity (from 0.88 to 0.98). Two inoculum concentrations were used (10 and 10(4) CFU g(-1)), and the growth limits for the two inocula were different. The kinetic behavior of L. monocytogenes as a function of temperature (4, 8, 12, and 16 degrees C) on the same meat product at the lower concentration (10 CFU g(-1)) was also studied. The Baranyi model appeared to fit the overall experimental data better than did the modified Gompertz and the modified logistic models. Maximum specific growth rate (micromax), lag phase duration (LPD), and maximum cell concentration (Nmax) derived from the primary model were modeled using the square root function (micromax and LPD) and a second order polynomial (Nmax) (secondary models). The selection of the best model (primary or secondary) was based on some statistical indices (the root mean square error of residuals of the model, the regression coefficient, the F test, the goodness of fit, and the bias and accuracy factor). The developed kinetic behavior model was validated under constant and dynamic temperature storage conditions. This prediction of L. monocytogenes growth provides useful information for improving meat safety and can be used for in-depth inspection of quality assurance systems in the meat industry.     

低温条件下即食虾中单增李斯特菌与副溶血性弧菌共存分子预测模型的建立

本研究旨在应用实时荧光定量聚合酶链式反应（quantitative real-time polymerase chain reaction,qPCR）技术描述即食虾中的单增李斯特菌与副溶血性弧菌的生长行为,构建混菌模式下的分子预测模型。将单增李斯特菌与副溶血性弧菌等量（（5.0±0.5）（lg（CFU/g）））混合接种于即食虾中,置于低温环境（4 ℃和10 ℃）下培养,并利用qPCR定量检测单增李斯特菌与副溶血性弧菌数量的动态变化。运用生长模型（修正Gompertz、Logistic、Baranyi）和失活模型（Log-linear、Weibull）分别拟合两株菌的生长和失活趋势。结果表明：低温条件下,修正Gompertz、Logistic和Baranyi模型均可成功拟合单增李斯特菌的生长曲线,其决定系数（R2）均大于0.98。对于副溶血性弧菌,在4 ℃条件下,Log-linear和Weibull模型能够清晰地描述其失活情况,R2分别为0.950和0.945;而10 ℃条件下,2 个失活模型均难以描述其行为变化,R2仅为0.784和0.775。本研究运用分子生物学技术描述即食虾中两种致病菌混合培养的菌量变化,探究混菌模式下微生物的生长失活情况,为分子预测模型的进一步研究提供了新的思路。     

Description of growth of Clostridium perfringens in cooked beef with multiple linear models

The traditional linear model used in food microbiology employs three linear segments to describe the process of food spoilage and categorize a growth curve into three phases — lag, exponential, and stationary. The linear model is accurate only within certain portions of each phase of a growth process, and can underestimate or overestimate the transitional phases. While sigmoid functions (such as the Gompertz and logistic equations) can be used to fit the experimental growth data more accurately, they fail to indicate the physiological state of bacterial growth. The objective of this paper was to develop a new methodology to describe and categorize accurately the bacterial growth as a process using Clostridium perfringens as a test organism. This methodology utilized five linear segments represented by five linear models to categorize a bacterial growth process into lag, first transitional, exponential, second transitional, and stationary phases. Growth curves described in this paper using multiple linear models were more accurate than the traditional three-segment linear models, and were statistically equivalent to the Gompertz models. With the growth rates of transitional phases set to 1/3 of the exponential phase, the durations of the lag, first transitional, exponential, and second transitional phases in a growth curve described by the new method were correlated linearly. Since this linear relationship was independent of temperature, a complete five-segment growth curve could be generated from the maximum growth rate and a known duration of the first four growth phases. Moreover, the lag phase duration defined by the new method was a linear function of the traditional lag phase duration calculated from the Gompertz equation. With this relationship, the two traditional parameters (lag phase and maximum growth rate) used in a three-segment linear model can be used to generate a more accurate five-segment linear growth curve without involving complicated mathematical calculations.     

A new logistic model for Escherichia coli growth at constant and dynamic temperatures

A new logistic model for bacterial growth was developed in this study. The model, which is based on the logistic model, contains an additional term for expression of the very low rate of growth during a lag phase, in its differential equation. The model successfully described sigmoidal growth curves of Escherichia coli at various initial cell concentrations and constant temperatures. The model predicted well the bacterial growth curves, similar to the Baranyi model and better than the modified Gompertz model, especially in terms of the rate constant and the lag period of the growth curves. Using the experimental data obtained at the constant temperatures, the new logistic model was studied for growth prediction at a dynamic temperature. The model accurately described E. coli growth curves at various patterns of dynamic temperature. It also well described other bacterial growth curves reported by other investigators. These results showed that this model could be a useful tool for bacterial growth prediction from the temperature history of a tested food.     

PREDICTIVE MODELING AND VALIDATION OF GROWTH AT DIFFERENT TEMPERATURES OF BROCHOTHRIX THERMOSPHACTA

The growth of Brochothrix thermosphacta affect by temperatures (0, 2, 5, 7 and 10C) were studied in laboratory medium. Growth curves were fitted using logistic, Gompertz and Baranyi models. Statistical characteristics like r ², mean square error, bias factor and accuracy factor were using for comparison of these models. Based on the criteria, the Gompertz described the data best, Baranyi performed the predicting best. The maximum growth rates obtained from primary model were then modeled as a function of temperature using the square root model. Statistically for the secondary model, the bias and accuracy factors are 0.9978, 0.9943 and 0.9712, and 1.0513, 1.0639 and 1.2225 for logistic, Gompertz and Baranyi, respectively, which may indicate that Baranyi model fitted and performed best of the others. The 95% confidence limits for a new prediction were estimated for each validation temperatures condition using a SPSS procedure.

PRACTICAL APPLICATIONS

Brochothrix thermosphacta was isolated from chilled pork used for growth modeling in broth (tryptone soya broth) at pH 7 and NaCl 0.5% (w/w), at various temperatures. Data obtained from experiments were then fitted by three types of primary models, to estimate the maximum growth rate. A secondary model was created, describing how the maximum growth rate responds to the temperatures. The goodness-of-fit and statistical characteristics of the equations were tested.     

真空包装狮子头货架期预测模型的建立

为建立真空包装狮子头货架期预测模型,分析不同温度贮藏期间狮子头中菌落总数的变化情况,分别用线性模型、修正的Gompertz模型、修正的Logistic模型和Baranyi模型对狮子头中菌落总数进行一级模型的拟合,在此基础上使用平方根模型建立二级模型。通过比较各模型的评价参数选择最优模型,并进一步建立货架期预测模型。结果表明在一级模型中,修正的Gompertz模型对真空包装狮子头中菌落总数生长曲线的拟合优度最高;基于修正的Gompertz模型建立的平方根模型可较好地描述温度对狮子头最大比生长速率和迟滞期的影响。在4、10、15、20、25℃条件下贮藏狮子头的货架期分别为80.79、45.22、10.96、4.96、4.01 d,货架期实测值与预测值的相对误差值均在10%以内,表明建立的模型可以较准确地对贮藏在4~25℃条件下的狮子头进行货架期预测。     

Comparison of Mathematical Models of Lactic Acid Bacteria Growth in Vacuum‐Packaged Raw Beef Stored at Different Temperatures

The lactic acid bacteria grown in vacuum‐packaged raw beef under 7, 10, 15, and 20 °C has been studied in this paper. Four primary models, the modified Gompertz, logistic, Baranyi, and Huang model were used for data fitting. Statistical criteria such as the bias factor and accuracy factor, mean square error, Akaike's information criterion, and the residual distribution were used for comparing the models. The result showed that all of the 4 models can fit the data well and they were not significantly different in the performance. They were equally capable of describing bacterial growth, but the growth rate and lag time estimated from the modified Gompertz model were a little higher than other models. The estimate for the lag time was not accurate as the growth rate.     

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。     

真空包装鸡肉早餐肠中细菌总数生长预测模型的拟合优度比较

为比较不同生长预测模型对真空包装鸡肉早餐肠中细菌总数生长情况的拟合效果,观察在不同贮藏温度（2～15 ℃）下,使用Baranyi、修正的Gompertz及修正的Logistic模型分别描述细菌总数随时间变化的情况,以及使用Arrhenius方程与平方根模型描述一级模型所得参数随温度变化的情况。通过计算各模型的评价参数（均方误差平方根RMSE、回归系数R2、赤池信息准则与贝叶斯信息准则）,参考模型所得特征值及货架期残差值,评价各模型的拟合优度,寻找最优组合。结果表明：Baranyi模型所得方程的评价参数最优,最大比生长速率（μmax）最大,所得产品货架期残差值较小;应用修正的Gompertz模型更有利于优化二级模型评价参数;而修正的Logistic模型拟合所得初始菌数N0值偏小,且将15 ℃贮藏组延滞时间λ计算为负值。因此Baranyi模型的拟合优度最高,其次为修正的Gompertz模型,最后为修正的Logistic模型。应用Arrhenius方程与平方根模型均能够成功拟合,但未能得出拟合更优者。     

A MODEL FOR PREDICTING THE GROWTH OF LISTERIA MONOCYTOGENES IN PACKAGED WHOLE MILK

Experiments were conducted to determine growth characteristics of Listeria monocytogenes in sterilized whole milk at nine temperatures in the range of 277.15 to 308.15K (4 to 35C). Based on these data, the parameter values of the Baranyi dynamic growth model were statistically determined. Finite element software, ANSYS, was used to determine temperature distributions in milk cartons subject to a time‐varying ambient temperature profile. The space‐time‐temperature data were input to the Baranyi dynamic growth model, to predict the microbial population density distribution and the average population density in the milk carton. The Baranyi dynamic growth model and the finite element model were integrated and validated using experimental results from inoculated sterilized whole milk in half‐gallon laminated paper cartons. In all experiments, the milk cartons were subjected to the same temperature profile as the Baranyi dynamic growth model. Experimental microbial counts were within predicted upper and lower bounds obtained using the integrated Baranyi dynamic growth and finite element models. In addition, the growth curve at the mean value of initial physiological state parameter for L. monocytogenes underpredicted the microbial growth (standard error = 0.54 log (cfu/mL) and maximum relative difference = 15.49%).     

Modelling of bacterial growth with shifts in temperature using automated methods with Listeria monocytogenes and Pseudomonas aeruginosa as examples

Time to detection (TTD) measurements using turbidometry allow a straightforward method for the measurement of bacterial growth rates under isothermal conditions. Growth rate measurements were carried out for Listeria monocytogenes at 25, 30 and 37°C and for Pseudomonas aeruginosa over the temperature range 25 to 45°C. The classical three-parameter logistic model was rearranged to provide the theoretical foundation for the observed TTD. A model was subsequently developed for the analysis of TTD data from non-isothermal studies based on the Malthusian approximation of the logistic model. The model was able to predict the TTD for cultures of L. monocytogenes or P. aeruginosa undergoing simple temperature shunts (e.g. 25 to 37°C and vice versa), and for a multiple temperature shunt for L. monocytogenes (25-37-25-37°C and 37-25-37-25°C) over a period of 24h. In no case did a temperature shunt induce a lag.