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.     

热死环丝菌生长预测模型的建立

针对引起冷却肉腐败的优势微生物热死环丝菌,以分离得到的S-8菌株作为受试菌株,研究了0～10℃低温条件下热死环丝菌S-8的生长情况。利用SAS程序拟和不同温度条件下的生长情况,经过比较发现,Gompertz模型比线性模型能更好地拟合热死环丝菌的生长,从而得到了其生长的Gompertz模型参数;利用平方根模型对其的最大比生长速率平方根-温度(U(1/2)-T)进行拟合,得到热死环丝菌S-8生长的二级模型:U(1/2)=0.1192(T+6.00),T∈[0,10];利用培养基数据,冷却肉产品数据和温度波动条件下的数据对所得到的二级模型进行验证,计算得到总的偏差因子和准确因子分别为0.982和1.223,结果表明二级模型能真实快速有效地预测冷链条件下冷却肉中热死环丝菌的生长。     

草鱼鱼整片中热杀索丝菌的生长预测模型

为了研究热杀索丝菌(Brochothrix thermosphacta)的生长动态,建立了5、10、15、20℃四种不同温度下草鱼鱼整片中的热杀索丝菌的预测模型.利用Gompertz方程获得热杀索丝菌的生长预测值,根据预测值和恒定温度下的活菌数,绘制实际和预测生长曲线,曲线重合度较好.利用平方根模型描述温度对最大比生长速率和延滞期的影响.通过计算准确因子和偏差因子对一级模型进行了验证,结果准确因子AF值均在1左右,偏差因子BF值在0.7～1.1之间.利用F统计量对二级模型进行验证,得到的P值小于0.05.得到的预测模型能很好的预测热杀索丝菌在草鱼鱼整片中的生长动态,为水产品的预报模型在实际生产和流通过程中的提供一定参考.     

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。     

冷鲜梅条肉中热杀索丝菌生长预测模型建立与检验

研究冷鲜梅条肉中热杀索丝菌在0℃、5℃、10℃、15℃、20℃不同温度下生长变化情况,利用Modified Gompertz模型建立热杀索丝菌一级生长预测模型(R2＞0.99);利用平方根模型描述温度与最大比生长速率和延滞期的关系,得到热杀索丝菌的生长预测二级模型,验证模型的数学参数准确因子Af、Bf在1左右.表明数学模型可用于预测0℃～20℃范围内热杀索丝菌的变化情况,为冷鲜肉的货架期预报提供了基础数据.     

冷鲜肉中热杀索丝菌生长预测模型的建立与验证

以市售托盘装冷鲜猪肉为研究对象测定热杀索丝菌的数量变化情况与感官、挥发性盐基氮和菌落总数的变化,结果表明冷鲜猪肉的腐败限控量为5.316 lg(CFU/g) ,热杀索丝菌在不同温度货架期终点时菌落数均值为7.519lg(CFU/g)。运用统计学软件SAS9.1 拟合热杀索丝菌在不同温度下的生长动力模型,表明Gompertz 模型能很好拟合热杀索丝菌在不同温度下的生长;利用平方根模型描述温度与最大比生长速率和延滞期的关系,得到热杀索丝菌生长的二级模型,判定系数R2 的值均在0.99 以上,表明温度与最大比生长速率和延滞期之间存在良好的线性关系;建立了0～15℃温度区域内冷鲜猪肉储藏过程中的货架期预测模型,用3℃储藏冷鲜肉中热杀索丝菌生长的实测值与通过货架期预测模型得到的预测值进行比较,相对误差为1.6%,表明模型可以可靠预测0～15℃温度区域内冷鲜猪肉的货架期。     

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。     

冷鲜肉馅中热杀索丝菌预测模型的建立

通过建立数学模型,来快速准确预测和监控瘦肥比例为7:3的冷鲜猪肉馅中优势腐败菌生长情况,选取0、5、10、15、20℃五种不同的温度,建立和验证0～20℃条件下热杀索丝菌的生长预测模型。结果表明,利用Gompertz模型拟合0～20℃条件下热杀索丝菌的生长,判定系数R2均大于0.99,计算得到总的偏差因子和准确因子分别为1.16和0.88;利用平方根模型描述了温度与最大比生长速率和延滞期的关系,二者都呈现了良好的线性关系;由此建立的0～20℃条件下7:3冷鲜猪肉馅中热杀索丝菌的生长预测模型有比较好的预测效果。     

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.     

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.     

Mathematical modeling on the growth of Staphylococcus aureus in sandwich

The growth of Staphylococcus aureus in sandwich fillings at different incubation temperatures was tested. These growth data were fitted into the Gompertz model, Logistic model, and Baranyi model in order to compare the goodness-of-fit of the 3 primary models using several factors such as coefficient of determination (R²), the standard deviation (S_y.x), and the Akaike’s information criterion (AIC). The Gompertz model showed the best statistical fit. Hence, growth parameters such as specific growth rate (SGR) and lag time (LT) obtained from the Gompertz model were used to construct the secondary models. Further, developed models were evaluated by bias factor (B_f) and accuracy factor (A_f). For the SGR, the Bf value was 0.993 and A_f value was 1.156 which indicated conservative predictions. While for LT, a clear deviation was observed between predictions and observations (B_f=0.635 and A_f=1.592). The results, however, were also considered acceptable after comparing with previous publications.     

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。     

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。     

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.     

Predictive Modeling for Growth of Non‐ and Cold‐adapted Listeria monocytogenes on Fresh‐cut Cantaloupe at Different Storage Temperatures

The aim of this study was to determine the growth kinetics of Listeria monocytogenes, with and without cold‐adaption, on fresh‐cut cantaloupe under different storage temperatures. Fresh‐cut samples, spot inoculated with a 4‐strain cocktail of L. monocytogenes (～3.2 log CFU/g), were exposed to constant storage temperatures held at 10, 15, 20, 25, or 30 °C. All growth curves of L. monocytogenes were fitted to the Baranyi, modified Gompertz, and Huang models. Regardless of conditions under which cells grew, the time needed to reach 5 log CFU/g decreased with the elevated storage temperature. Experimental results showed that there were no significant differences (P > 0.05) in the maximum growth rate k (log CFU/g h^?1) and lag phase duration λ (h) between the cultures of L. monocytogenes with or without previous cold‐adaption treatments. No distinct difference was observed in the growth pattern among 3 primary models at various storage temperatures. The growth curves of secondary modeling were fitted on an Arrhenius‐type model for describing the relationship between k and temperature of the L. monocytogenes on fresh‐cut cantaloupe from 10 to 30 °C. The root mean square error values of secondary models for non‐ and cold‐adapted cells were 0.018, 0.021, and 0.024, and 0.039, 0.026, and 0.017 at the modified Gompertz, Baranyi, and Huang model, respectively, indicating that these 3 models presented the good statistical fit. This study may provide valuable information to predict the growth of L. monocytogenes on fresh‐cut cantaloupes at different storage conditions.     

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。     

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.     

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.     

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.     

Analysis and mathematical modelling of the behaviour of Escherichia coli in the mascarpone cheese during cold storage

The aim of this work was to study the behaviour of E. coli in mascarpone cheese during storage at the temperatures ranging from 3 to 15 °C, as well as application of predictive microbiology to describe the experimental data. The Baranyi, Gompertz and logistic models were fitted at the stage of primary modelling. Although all applied primary models described the growth of micro‐organisms accurately, the most accurate goodness of fit was obtained for the Gompertz model and the growth rates generated by this model were used for secondary modelling. The polynomial model predicted accurately the influence of temperature on the growth rate of E. coli, reaching the adjusted coefficient of linear regression 0.99. Generated predictive model that describes the growth of E. coli in mascarpone cheese constitutes a valuable tool in assessing the microbiological stability of the food product with similar physicochemical properties.