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.     

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.     

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.     

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。     

Modeling survival of high hydrostatic pressure treated stationary- and exponential-phase Listeria innocua cells

High Hydrostatic Pressure (HHP) inactivation (325–400 MPa; 0–20 min; maximum temperature 30 °C) of cells of Listeria innocua CECT 910 was studied in two different growth phases (exponential and stationary), and the corresponding survival curves were obtained for each case. The curves were fitted to two nonlinear models, the modified Gompertz equation and the Baranyi model. The kinetic constants calculated for both models, µ_max and k_max, indicated that cells in exponential growth phase were more sensitive to pressure than those in stationary phase. Both mathematical models were suitable for describing L. innocua HHP survival curves, rendering kinetic constants that increased with increasing pressure. When considering the experimental models validation, both Gompertz and Baranyi predicted in a similar way, however Baranyi had slightly lower A_f (Accuracy factor) and B_f (Bias factor) values, which indicated better prediction values. In summary, both mathematical models were perfectly valid for describing L. innocua inactivation kinetics under HHP treatment.Industrial relevanceThe mathematical models for inactivation and growth of microorganisms are the foundation of predictive microbiology and are used in risk assessments procedures as part of the food safety management system. Besides, these models together with those applied to inactivation of enzymes and destruction of quality factors are essential to optimize processes and thus to lay the foundations for industrial processing. It is therefore necessary to identify generally applicable kinetic models that will produce primary and secondary kinetic parameters and are statistically reliable as a key tool to predict the behaviour of microorganisms, enzymes and quality factors after processing.     

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.     

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.     

Analysis of mathematical models of Pseudomonas spp. growth in pallet-package pork stored at different temperatures

Pseudomonas of pallet-packaged raw pork grown at 0, 5, 10, 15, 20 and 25 °C has been studied in this paper. The modified Gompertz, Baranyi and Huang models were used for data fitting. Statistical criteria such as residual sum of squares, mean square error, Akaike's information criterion, and pseudo-R² were used to evaluate model performance. Results showed that there was an apparent decline in Pseudomonas growth at initial-storage phase at low temperatures. The modified Gompertz model outperformed the others at 5, 15, and 20 °C, while Baranyi model was appropriate for 0 and 25 °C. The Huang model was optimal at 10 °C. No single model can give a consistently preferable goodness-of-fit for all growth data. The Gompertz model, with the smallest average values of RSS, AIC, MSE and the biggest pseudo-R² at all temperatures, is the most appropriate model to describe the growth of Pseudomonas of raw pork under pallet packaging.     

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。     

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.     

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。     

Pressure inactivation kinetics of Enterobacter sakazakii in infant formula milk

Survival curves of Enterobacter sakazakii inactivated by high hydrostatic pressure were obtained at four pressure levels (250, 300, 350, and 400 MPa), at temperatures below 30 degrees C, in buffered peptone water (BPW; 0.3%, wt/vol) and infant formula milk (IFM; 16%, wt/vol). A linear model and four nonlinear models (Weibull, log-logistic, modified Gompertz, and Baranyi) were fitted to the data, and the performances of the models were compared. The linear regression model for the survival curves in BPW and IFM at 250 MPa has fitted regression coefficient (R2) values of 0.940 to 0.700, respectively, and root mean square errors (RMSEs) of 0.770 to 0.370. For the other pressure levels, the linear regression function was not appropriate, as there was a strong curvature in the plotted data. The nonlinear regression models with the log-logistic and modified Gompertz equations had R2 values of 0.960 to 0.992 and RMSE values of 0.020 to 0.130 within pressure levels of 250 to 400 MPa, respectively. These results indicate that they are both better models for describing the pressure inactivation kinetics of E. sakazakii in IFM and BPW than the Weibull distribution function, which has an R2 minimum value of 0.832 and an RMSE maximum value of 0.650 at 400 MPa. On the other hand, the parameters for the Weibull distribution function, log-logistic model, and modified Gompertz equation did not have a clear dependence on pressure. The Baranyi model was also analyzed, and it was concluded that this model provided a reasonably good fit and could be used to develop predictions of survival data at pressures other than the experimental pressure levels in the range studied. The results provide accurate predictions of survival curves at different pressure levels and will be beneficial to the food industry in selecting optimum combinations of pressure and time to obtain desired target levels of E. sakazakii inactivation in IFM.     

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。     

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.     

Modelling the effect of environmental factors on the growth of Aspergillus parasiticus and mycotoxin production in paddy during storage

The simulated experiment of A. parasiticus isolated from the paddy was carried out during the paddy storage for 20 days. The growth and mycotoxin data were collected for constructing kinetic and probability models of moulds. The Baranyi and Gompertz model was employed as the primary model and estimated the lag phase and maximum specific growth rate. Secondary models, such as polynomial, Davey and Gibson model were used and completely evaluated under different conditions. The polynomial equation was highly rated compared with Gibson and Davey model and gave realistic temperatures and a_w for mould growth. Logistic model showed promising results on the prediction of growth boundary and AFB₁ production. Employed models showed promising predicted results, indicating that it is an effective tool for describing and predicting the growth of moulds under different temperatures and a_w. The results can be applied to develop the optimal strategy to prevent fungal spoilage and aflatoxin production during paddy storage.     

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。     

Modeling the inactivation of Salmonella typhimurium by dense phase carbon dioxide in carrot juice

The inactivation of Salmonella typhimurium inoculated into acidified carrot juice subjected to dense phase carbon dioxide (DPCD) was investigated. The pressures in the study were 10, 20 and 30 MPa, the temperatures were 32, 37 and 42 °C, and the treatment time was 5–90 min. The inactivation effect of DPCD was enhanced by increasing pressure and temperature. The sigmoid inactivation curves were characterized with the lag phase, exponential inactivation phase, and resistant phase. The inactivation curves were fitted to the modified Gompertz equation and the modified Logistic equation, the modified Gompertz equation was superior since its lowest residual sum of squares (RSS) was lower although there was no significant difference of goodness-of-fit between both models as indicated by F-test. The λ (the duration of the lag phase) and t_4-D (the time necessary to achieve 4-log cycles reduction) decreased with increasing pressure or temperature. The k_dm (the maximum specific value of the inactivation rate, min⁻¹) increased with increasing temperatures, and decreased with increasing pressures. The activation energy (Ea) and the activation volume (Va) necessary for inactivating S. typhimurium by DPCD were 19.06–29.39 kJ mol⁻¹ and 18.89–58.27 cm³ mol⁻¹.     

不同温度下腐败希瓦氏菌（Shewanella putrefaciens）生长动力学模型的比较与评价

以冷藏大黄鱼特定腐败菌腐败希瓦氏菌（Shewanella putrefaciens）为研究对象,采用修正Gompertz、修正Logistic和Baranyi方程拟合5、8、15 ℃和25 ℃条件下其在胰蛋白胨大豆肉汤中的生长动力学模型,采用Belehradek方程建立二级模型,探讨温度对腐败希瓦氏菌生长动力学的影响,并对模型的拟合优度及适用性进行评价。结果表明：温度对腐败希瓦氏菌生长动力学影响显著,其在5 ℃环境中延滞期较长,生长趋势得到明显抑制,当温度上升到25 ℃时,腐败希瓦氏菌的延滞期显著缩短,比生长速率随着温度的升高而增大,温度与延滞期及比生长速率均存在线性关系。采用均方根误差（root mean square error,RMSE）、残差平方和（residual sumof squares,RSS）、偏差度（bias factor,BF）、准确度（accuracy factor,AF）、R2对修正的Gompertz、修正的Logistic和Baranyi方程的拟合优度进行评价,修正的Logistic方程的RSS和RMSE均最小,BF和AF均最接近1,修正的Logistic模型的拟合优度最佳,适用性最强,水产品中腐败希瓦氏菌的生长情况能通过修正的Logistic模型得到较好地预测。     

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.     

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.