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.     

Development of predictive model for the growth of Staphylococcus aureus in Kimbab

This study was conducted to develop predictive models for the growth of Staphylococcus aureus in kimbab as a function of storage temperatures (7, 10, 12, 14, 16, 20, 25, and 30°C). The growth data were fitted into the modified Gompertz model and the Logistic model, and the goodness-of-fit of primary models was compared using determination of coefficient, mean square error, and Akaike’s information criterion. The modified Gompertz model was found to be more suitable to describe the growth data. Therefore, the growth rate (GR) and lag time (LT) obtained from the modified Gompertz model were employed to establish the secondary models. The newly developed models were validated using root mean square error (RMSE), bias factor (B_f), and accuracy factor (A_f). The results showed that RMSE<0.20 and B_f and A_f values were within the reliable range, which indicated that the presented predictive models can be used to assess the risk of S. aureus infection in kimbab.     

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.     

Development and validation of growth model for Yersinia enterocolitica in cooked chicken meats packaged under various atmosphere packaging and stored at different temperatures.

Mathematical models that can predict the growth of Yersinia enterocolitica in chicken meats were evaluated in this study. The growth curves for Y. enterocolitica in chicken meats variously packaged (air, vacuum, and modified atmosphere packaging [MAP]) and stored at various temperatures (4, 10, 16, 22, 28, and 34 degrees C) were constructed. The Gompertz model was applied to fit each of the experimental curves for the conditions mentioned above. The variations in the parameters, including lag time (lambda) and specific growth rate (mu), at various temperatures were then described by the following models: the variations in lag time were described by the Adair and Smith models and the variations in the specific growth rate were described by the Ratkowsky and Zwietering models. The various models were then compared using graphical and mathematical analyses such as mean square error (MSE), regression coefficient (r2), bias factor, and accuracy factor. The results indicate that the mean r values in the Gompertz model for chicken meats packaged in air, vacuum, and MAP were 0.99, 0.99, and 0.95, respectively. The lag time modeled with the Adair and Smith functions exhibited a greater variance and demonstrated larger errors. The MSEs were 0.0015 and 0.0017 for Ratkowsky and Zwietering models, respectively. The r2 values in the Ratkowsky and Zwietering models were both 0.99. The bias factor was 1.017 for the Ratkowsky model and 1.096 for the Zwietering model. The accuracy factor of the Zwietering model was 1.174, which was lower than that in the Ratkowsky model (1.275), indicating that the former model was more accurate than the latter in predicting the specific growth rate of Y. enterocolitica in chicken meats.     

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。     

铜绿假单胞菌在牛乳中生长的预测

以食源致病性铜绿假单胞菌ATCC27853为试材,研究其在牛乳中的生长模型参数与温度之间的关系,为其在牛乳中的安全控制提供理论依据。将ATCC27853接种于新鲜灭菌的牛乳中,分别置于6、10、16、22、28、36、42、45、48、50 ℃共10 个温度下生长,采用Matlab软件建立了在不同温度下ATCC27853的一级Gompertz模型;基于Gompertz模型拟合的参数,结合修正Ratkowsky模型与Hyperbola模型,分别建立了ATCC27853的最大比生长速率（μmax）与温度、延滞时间（λ）与温度之间的二级模型;采用决定系数、均方根误差、准确因子和偏差因子对ATCC27853的一级和二级模型进行评价。对一级Gompertz模型和二级修正Ratkowsky模型、Hyperbola模型进行验证,结果表明：一级Gompertz模型显著,能较好地预测不同温度下ATCC27853在牛乳中的生长;二级修正Ratkowsky模型和Hyperbola模型均显著,且拟合度较好;在6、10、48 ℃时,初始生理状态参数（h0）明显高于16～45 ℃时。ATCC27853的生长温度对μmax、λ和h0的影响可用于其在牛乳的加工、运输、贮藏和销售等过程中的安全预测,为ATCC27853在牛乳中的安全控制提供理论依据。     

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.     

不同温度条件下三文鱼中荧光假单胞菌生长预测模型的建立

荧光假单胞菌SBW25(Pseudomonas fluorescence SBW25)作为三文鱼的特定腐败菌(specific spoilage organisms,SSOs),在三文鱼腐败过程中起着主导作用。实验选择以三文鱼的鱼肉和鱼汁为生长介质,采用修正的Gompertz方程和Belehradek方程拟合三文鱼特定腐败菌之一的荧光假单胞菌SBW25在不同温度条件下的生长动力学模型,同时探究鱼汁中蛋白酶活力对动力学参数的影响,并对模型的适用性进行评价。结果显示,修正的Gompertz方程所拟合出的各温度下货架期方程的决定系数(R~2)都达到0.999,适用于描述三文鱼鱼肉和鱼汁中微生物的生长曲线。随着温度升高,鱼肉和鱼汁中的荧光假单胞菌的最大比生长速率和延滞期出现上升和缩短。鱼汁中温度和最大比生长速率、延滞期的Belehradek平方根方程取得较高的决定系数,分别达到0.930 3和0.988 7,高于鱼肉中取得的。在鱼汁中蛋白酶活力和对应时期荧光假单胞菌的最大比生长速率出现相同变化趋势。基于Belehradek方程的鱼汁不同温度模型偏差度和准确度都更接近于1.00,说明鱼汁中SSOs的生长曲线能较好地反映出各温度条件下的三文鱼货架期。     

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.     

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.     

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.     

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。     

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。     

Analysis of a molecular predictive mode for the growth of Staphylococcus aureus in pork

The objective of this study was to develop a molecular predictive model from quantitative real-time polymerase chain reaction methods to describe the growth of S. aureus strains in arti?cially contaminated pork in storage dependent of a constant temperature (7–30°C). This model acquired by quantitative real-time polymerase chain reaction methods was compared to a conventional predictive model using data. This study used three of the main growth models to fit the growth equation. The results proved that Modified Gompertz, Logistic, and Richards models were adequate for describing the growth curves. These models had the very low rate of the growth of S. aureus in pork during a lag phase. The growth rate increased with temperature, and the lag time decreased. Lag phases were apparent in all models, and those samples stored at low temperatures had longer lag phases. There was no significant difference in the molecular and conventional predictive models for any of the growth curves. However, the use of a molecular predictive model could save more time and labor to construct more precise models of certain pathogens. In conclusion, the molecular predictive model could provide an effective method to lessen the risk of S. aureus of pork.     

Modeling the response of Listeria monocytogenes at various storage temperatures in pork with/without electrolyzed water treatment

The objective of this study was to develop a model of the growth of Listeria monocytogenes in pork untreated or treated with low concentration electrolyzed water (LcEW) and strong acid electrolyzed water (SAEW), as a function of temperature. The experimental data obtained under different temperatures (4, 10, 15, 20, 25, and 30°C) were fitted into the modified Gompertz model to generate the growth parameters including specific growth rate (SGR) and lag time (LT) with high coefficients of determination (R² >0.97). The obtained SGR and LT were employed to develop square root models to evaluate the effects of storage temperature on the growth kinetics of L. monocytogenes in pork. The values of bias factor (0.924–1.009) and accuracy factor (1.105–1.186), which were regarded as acceptable, demonstrated that the obtained models could provide good and reliable predictions and be suitable for the purpose of microbiological risk assessment of L. monocytogenes in pork.     

Evaluation of primary models to describe the growth of Pichia anomala and study of temperature, NaCl, and pH effects on its biological parameters by response surface methodology

Tolerance of Pichia anomala, a strain of yeast associated with olive fermentation, to salt, temperature, and pH was studied in yeast-malt-peptone-glucose medium using a nonfactorial central composite experimental design with three repetitions in the center to account for pure error. Modified Gompertz, logistic, Richards-Stannard, and Baranyi-Roberts models were used to determine maximum specific growth rate (micro(max)) and lag phase period (lambda) from the growth curves (primary models). All models produced a good fit (significant at P < 0.05), but the graphical and statistical analyses of the data indicated that the modified Gompertz and Richards-Stannard models were the most appropriate. The biological parameters obtained with the diverse models were fitted to a response surface secondary model. A significant decrease in micro(max) was observed as temperature decreased and salt increased. A significant increase in lambda was observed as temperature (linear and quadratic effects) and pH decreased and as salt content increased. Effects of interactions were complex and depended on models. Validation revealed acceptable errors and bias in micro(max) and lambda values obtained in independent experiments. Validation growth curves were best reproduced by using the values of micro(max) and lambda predicted by the response surface from the modified Gompertz and Richards-Stannard models. Results from this study can be applied to table olive fermentation or storage and for production of table olives as refrigerated commercial products without the use of preservatives or pasteurization.     

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.     

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.     

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.     

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。