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.     

Characteristics and modeling of Escherichia coli growth in pouched food

Characteristics of the growth kinetics of Escherichia coli cells in pouched mashed potatoes under various conditions were studied with a mathematical model. Bacterial cells were inoculated in sterile mashed potatoes and then sealed in vinyl pouches, in which a very small amount of air was included. The growth curves of cells in the pouched mashed potatoes at constant temperature (12-34 degrees C) were sigmoidal with time on a semi-logarithmic plot and were successfully described with a new logistic model recently developed by us. The rate constant of growth showed a highly linear relationship to the temperature with the square-root model, and the lag period was longer at lower temperatures. The growth curve in glass tubes containing a large volume of air was similar to that in pouches, showing that the rate of growth was not affected by the volume of the surrounding air. The growth curves in pouched mashed potatoes were very similar to those in nutrient broth or on the surface of nutrient agar, which we previously reported. These results suggested that the growth kinetics of the bacterial cells under various conditions of rich nutrition might be almost identical, and can be described with a simple growth model like ours.     

Improvement of new logistic model for bacterial growth

Recently Fujikawa et al. [J. Food Hyg. Soc. Japan, 44, 155-160 (2003)] developed a new logistic model for bacterial growth. In the present study, an adjustment factor in the model was improved. The improved model could successfully describe growth curves of Escherichia coli and Salmonella in liquid media. In particular, the model could describe the linear growth at the early logarithmic phase more accurately than the previous model, being similar in this respect to the Baranyi model. However, the improved model more accurately predicted the rate constant of growth and the duration of the lag time as compared with the Baranyi model. These results showed that the improved model has the potential to successfully predict microbial growth.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

A new logistic model for bacterial growth

A new logistic model for bacterial growth was developed in this study. The model is based on a logistic model, which is often applied for biological and ecological population kinetics. The new model is described by a differential equation and contains an additional term for suppression of the growth rate during the lag phase, compared with the original logistic equation. The new model successfully described sigmoidal growth curves of Escherichia coli and Salmonella under various initial conditions. Data for E. coli were obtained from our experiments and data for Salmonella from the literature. When the new model was compared with a modified Gompertz model, which is widely used by many predictive microbiology researchers, it proved to be superior to the Gompertz model. Further, Salmonella growth at varying temperature could be well simulated by the new model. These results indicate that the new model will be a useful tool to predict bacterial growth under various temperature profiles.     

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.     

乙型副伤寒沙门氏菌在鲜切黄瓜中的生长行为及预测模型建立

为建立不同温度条件下鲜切黄瓜中乙型副伤寒沙门氏菌的生长预测模型,将新鲜黄瓜切丁,添加乙型副伤寒沙门氏菌,分别在10、15、20、25、30和35℃下的恒温条件下贮藏,以观察细菌的生长。使用USDA综合病原体建模程序(USDA-IPMP)拟合每个温度下每种细菌的生长曲线,以找出描述该细菌生长的最适初级生长模型,并拟合得到最大比生长速率。通过温度对初级模型中最大比生长速率的生长动力学拟合,分别建立Ratkowsky、Huang rate、Cardinal、Arrhenius-type二级生长模型,并进行数学评估和实测样品验证。结果表明,实验数据和生长曲线显示乙型副伤寒沙门氏菌的生长表现出三个阶段,包括延滞期,指数期和稳定期。乙型副伤寒沙门氏菌的延滞期时间随着孵育时间的增加而降低。相反,乙型副伤寒沙门氏菌的生长速率随着孵育温度而增加,由此表明风险随温度的升高而增加。使用Baranyi和Huang初级模型分析两种病原体的生长曲线,使用Ratkowsky、Huang平方根模型、Cardinal和Arrhenius模型描述温度对贮藏时间细菌生长的影响,同时应用实验数据和样品实测验证评估所建立的预测模型。从该研究中获得的结果和预测模型可用于预测鲜切黄瓜产品中乙型副伤寒沙门氏菌的生长。     

Analysis of a novel class of predictive microbial growth models and application to coculture growth

In this paper, a novel class of microbial growth models is analysed. In contrast with the currently used logistic type models (e.g., the model of 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]), the novel model class, presented in Van Impe et al. (Van Impe, J.F., Poschet, F., Geeraerd, A.H., Vereecken, K.M., 2004. Towards a novel class of predictive microbial growth models. International Journal of Food Microbiology, this issue), explicitly incorporates nutrient exhaustion and/or metabolic waste product effects inducing stationary phase behaviour. As such, these novel model types can be extended in a natural way towards microbial interactions in cocultures and microbial growth in structured foods. Two illustrative case studies of the novel model types are thoroughly analysed and compared to the widely used model of Baranyi and Roberts. In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition and is described as a function of the pH-evolution. In the second case study, substrate exhaustion is the sole cause of the stationary phase. Finally, a more complex case study of a so-called P-model is presented, dealing with a coculture inhibition of Listeria innocua mediated by lactic acid production of Lactococcus lactis.     

Modelling the growth parameters of lactic acid bacteria and total viable count in vacuum‐packaged Irish cooked sausages cooled by different methods

The growth kinetics of lactic acid bacteria (LAB) and total viable count (TVC) in cooled sausages as a function of storage time was studied. Cooked sausages were cooled either by commercial cooling or by immersion vacuum cooling (IVC), then vacuum‐packaged and stored for up to 71 days at 4 °C. Baranyi model was used to fit the growth of LAB and TVC. Growth curve, growth rate, lag time and initial and final cell concentrations for LAB and TVC were calculated through DMFit. The coefficient of determination and root‐mean‐squared error of the Baranyi models were used to evaluate its accuracy. Sausages cooled by IVC had a longer lag time and shelf life period than commercial cooled sausages. Accuracy analysis showed that the Baranyi model was adequate for representing the bacterial growth of vacuum‐packaged cooled sausages and had a potential for shelf life prediction.     

Development of a Predictive Model for Spoilage of Cooked Cured Meat Products and Its Validation Under Constant and Dynamic Temperature Storage Conditions

ABSTRACT:  In the present study, the spoilage flora of a sliced cooked cured meat product was studied to determine the specific spoilage organism (SSO). The physicochemical changes of the product during its storage in a temperature range of 0 to 12 °C were also studied. Among the primary models used to model the temperature effect on SSO growth, the modified Gompertz described better the experimental data than modified logistic and Baranyi. The derived growth kinetic parameters, such as maximum specific growth rate (μ_max) and lag phase duration (LPD), were modeled by using the square root and Arrhenius equation (secondary models). The latter described better the data of μ_max and LPD; therefore, this model was chosen for correlating temperature with kinetic parameters. 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 coefficient of multiple determination, the F -test, the goodness of fit, the bias, and accuracy factor). The validation of the developed model was carried out under constant and dynamic temperature storage conditions. To validate its usefulness to similar products, another sliced cooked cured meat product stored under constant temperature conditions was also used. The log shelf life model was used for shelf life predictions based on the evident (visual defects) or the incipient spoilage (attainment of a certain spoilage level by SSO and/or chemical spoilage index). The possibility for shelf life predictions constitutes a valuable information source for the quality assurance systems of meat industries.     

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.     

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。     

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。     

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.