Indices for performance evaluation of predictive models in food microbiology

Two complementary measures are proposed as simple indices of the performance of models in predictive food microbiology. The indices assess the level of confidence one can have in the predictions of the model and whether the model displays any bias which could lead to 'fail-dangerous'predictions. The use of the indices is demonstrated using data collated from independent and published literature. This analysis supports previous reports that evaluation of predictive models by comparison to published microbial growth rate data may be inappropriate because of limitations in that data. The indices may fail to reveal some forms of systematic deviation between observed and predicted behaviour. It is concluded, however, that the indices provide an objective and readily interpreted summary of model performance and may serve as a first step towards the development of an objective and useful definition of the term 'validated model'in predictive food microbiology.     

Bělehrádek-type models

Summary Square-root (or Ratkowsky) models are a special case of Blehrádek's temperature rate-relationship first published in 1926 and widely used in several fields of biology. Blehrádek-type models also describe microbial growth, and have been extended for use in food microbiology by the inclusion of terms for water activity and pH. The parameters of the square root-type models are defined and their determination described. Favorable features of square root-type models include parsimony, parameter estimation properties, and ease of use. Square root-type models have been developed for a number of organisms of concern to the food industry and have also been adopted for use in a number of electronic devices used in predictive microbiology. Criticisms of square root-type models are also considered.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Modelling the bacterial growth/no growth interface

A logistic regression model is proposed which enables one to model the boundary between growth and no growth for bacterial strains in the presence of one or more growth controlling factors such as temperature, pH and additives such as salt and sodium nitrite. The form of the expression containing the growth limiting factors may be suggested by a kinetic model, while the response at a given combination of factors may either be presence/absence (i.e. growth/no growth) or probabilistic (i.e. r successes in n trials). The approach described represents an integration of the probability and kinetic aspects of predictive microbiology, and a unification of predictive microbiology and the hurdle concept. The model is illustrated using data for Shigella flexneri.     

Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

Predictive model for growth of <Emphasis Type="Italic">Listeria monocytogenes</Emphasis> in untreated and treated lettuce with alkaline electrolyzed water

This study was performed to develop predictive models for the growth kinetics of Listeria monocytogenes in Ready-to-Eat (RTE) lettuce treated with or without alkaline electrolyzed water. Firstly, growth curves of L. monocytogenes in treated and untreated RTE lettuce were obtained at several isothermal conditions (4, 10, 15, 20, 25, 30, and 35°C) and were then fitted into Gompertz model with a high correlation coefficient (R ² > 0.99). Growth parameters such as growth rate (GR) and lag time (LT) estimated by Gompertz model were found mostly have significant difference (P < 0.05) with those predicted by Combined database for predictive microbiology (ComBase). Moreover, increased GR and decreased LT were observed with increasing storage temperatures from 4 to 35°C and untreated lettuce showed lowest GR or longest LT, and followed by treated lettuce and ComBase, respectively. Furthermore, square root equation was employed to establish the secondary models for the GR to evaluate the effect of different storage temperatures on the growth rate of L. monocytogenes in untreated lettuce and treated lettuce. After that, verification of the developed models has been carried out using several mathematical or statistical indicators such as R ², the average mean square error (MSE), bias factor (B _f) and accuracy factor (A _f). It showed that R ² values were close to 1 (>0.95), and MSE calculated from models of untreated and treated lettuce were 0.0011 and 0.0008, respectively. Also, B _f values of 0.980 and 1.034 and A _f values of 1.107 and 1.118 were all in the acceptable range. This demonstrated that overall predictions showed good agreement with the experimental values, indicating success at providing reliable predictions of L. monocytogenes growth in RTE lettuce.     

食品微生物生长预测模型研究新进展

为了更好的了解食品微生物学预测模型的基本内容, 探讨数学模型在预测微生物学中的作用, 达到提高食品卫生检测效率, 保证食品质量安全的目的, 本文以文献综述形式, 简要概述了预测微生物学一级、二级和三级模型的含义与内容。并在此基础上, 着重介绍了全球范围内已经成功推广使用的多种三级模型, 阐述了它们的研究背景、研究进展、使用方法, 分析了各种模型存在的优缺点, 可为实际应用中选择合适的模型提供参考。在比较使用不同种类模型后, 发现Baranyi & Roberts、响应面和ComBase模型在各级模型中具有更好的使用价值。     

食品微生物生长预测模型研究新进展

为了更好的了解食品微生物学预测模型的基本内容,探讨数学模型在预测微生物学中的作用,达到提高食品卫生检测效率,保证食品质量安全的目的,本文以文献综述形式,简要概述了预测微生物学一级、二级和三级模型的含义与内容。并在此基础上,着重介绍了全球范围内已经成功推广使用的多种三级模型,阐述了它们的研究背景、研究进展、使用方法,分析了各种模型存在的优缺点,可为实际应用中选择合适的模型提供参考。在比较使用不同种类模型后,发现Baranyi＆Roberts、响应面和ComBase模型在各级模型中具有更好的使用价值。     

Modelling the growth,survival and death of Listeria monocytogenes

In this paper, the predictive microbiology approach has been generalized to the study of growth, survival and death of Listeria monocytogenes. As this micro-organism is involved in food poisoning, its growth, survival and death were studied as functions of low temperatures, NaCl and phenol compounds, in a synthetic medium, by a factorially designed experiment. A significant inactivation of L. monocytogenes was obtained with 20 ppm of phenol and 4% (w/v) NaCl at temperatures from 4 to 12 degrees C. An empirical model is proposed to describe, in a single step, the biomass profile vs studied factors. Thereby, the influence of temperature, NaCl and phenol concentration on L. monocytogenes biomass quantity (0.5-8 log cfu ml(-1)) are presented as a function of storage duration. The comparisons of the proposed model with existing models (Gompertz for growth, vitalistic for survival and death) were performed. The use of a single equation allows the prediction of contamination levels in all experimental conditions without knowledge a priori. The model offers considerable prospects for its use in food microbiology.     

Predictive modeling of mixed microbial populations in food products: evaluation of two-species models

Predictive microbiology is an emerging research domain in which biological and mathematical knowledge is combined to develop models for the prediction of microbial proliferation in foods. To provide accurate predictions, models must incorporate essential factors controlling microbial growth. Current models often take into account environmental conditions such as temperature, pH and water activity. One factor which has not been included in many models is the influence of a background microflora, which brings along microbial interactions. The present research explores the potential of autonomous continuous-time/two-species models to describe mixed population growth in foods. A set of four basic requirements, which a model should satisfy to be of use for this particular application, is specified. Further, a number of models originating from research fields outside predictive microbiology, but all dealing with interacting species, are evaluated with respect to the formulated model requirements by means of both graphical and analytical techniques. The analysis reveals that of the investigated models, the classical Lotka-Volterra model for two species in competition and several extensions of this model fulfill three of the four requirements. However, none of the models is in agreement with all requirements. Moreover, from the analytical approach, it is clear that the development of a model satisfying all requirements, within a framework of two autonomous differential equations, is not straightforward. Therefore, a novel prototype model structure, extending the Lotka-Volterra model with two differential equations describing two additional state variables, is proposed to describe mixed microbial populations in foods.     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

A prototype model structure for mixed microbial populations in homogeneous food products

An important factor which has not been included in many models in the field of predictive microbiology is the influence of a background of microflora in a food product. It is however generally known that the growth of a microorganism as a pure culture can be substantially different from its growth in a mixed culture, due to microbial interactions. Because of the importance of these interactions and the lack of suitable modeling techniques in the field of predictive microbiology to describe them, the potential of models in other research fields-namely ecology-to deal with interactions is explored in previous work of the authors. However, a model structure for microbial growth in food products cannot simply be copied from those elaborated in ecology. The structure of a predictive growth model is indeed typical, primarily due to the explicit modeling of a lag phase. The current paper proposes a prototype model structure for growth of mixed microbial populations in homogeneous food products. The model is able to describe a lag phase and reduces to a classical predictive growth model in the special case of single-species growth.     

Comparison of Modeling Methods for DoE‐Based Holistic Upstream Process Characterization

Upstream bioprocess characterization and optimization are time and resource‐intensive tasks. Regularly in the biopharmaceutical industry, statistical design of experiments (DoE) in combination with response surface models (RSMs) are used, neglecting the process trajectories and dynamics. Generating process understanding with time‐resolved, dynamic process models allows to understand the impact of temporal deviations, production dynamics, and provides a better understanding of the process variations that stem from the biological subsystem. The authors propose to use DoE studies in combination with hybrid modeling for process characterization. This approach is showcased on Escherichia coli fed‐batch cultivations at the 20L scale, evaluating the impact of three critical process parameters. The performance of a hybrid model is compared to a pure data‐driven model and the widely adopted RSM of the process endpoints. Further, the performance of the time‐resolved models to simultaneously predict biomass and titer is evaluated. The superior behavior of the hybrid model compared to the pure black‐box approaches for process characterization is presented. The evaluation considers important criteria, such as the prediction accuracy of the biomass and titer endpoints as well as the time‐resolved trajectories. This showcases the high potential of hybrid models for soft‐sensing and model predictive control.     

Evaluation of Data Transformations and Validation of a Model for the Effect of Temperature on Bacterial Growth

The temperature of chilled foods is an important variable for controlling microbial growth in a production and distribution chain. Therefore, it is essential to model growth as a function of temperature in order to predict the number of organisms as a function of temperature and time. This article deals with the correct variance-stabilizing transformation of the growth parameters A (asymptotic level), μ (specific growth rate), and λ (lag time). This is of importance for the regression analysis of the data. A previously gathered data set and model for the effect of temperature on the growth of Lactobacillus plantarum (M. H. Zwietering, J. T. de Koos, B. E. Hasenack, J. C. de Wit, and K. van 't Riet, Appl. Environ. Microbiol. 57:1094-1101, 1991) is extended with new data. With the total data set (original and new data), a variance-stabilizing transformation is selected in order to determine which transformation should precede fitting. No transformation for the asymptote data, a square root for the growth rate, and a logarithmic transformation for the lag time were found to be appropriate. After these transformations, no significant correlation was found between the variance and the magnitude of the variable. Model corrections were made and model parameters were estimated by using the original data. With the new data, the models were validated by comparing the lack of fit of the models with the measurement error, using an F test. The predictions of the models for μ and λ were adequate. The model for A showed a systematic deviation, and therefore a new model for A is proposed.     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.     

Using functional traits to predict species growth trajectories,and cross‐validation to evaluate these models for ecological prediction

Modeling plant growth using functional traits is important for understanding the mechanisms that underpin growth and for predicting new situations. We use three data sets on plant height over time and two validation methods—in‐sample model fit and leave‐one‐species‐out cross‐validation—to evaluate non‐linear growth model predictive performance based on functional traits. In‐sample measures of model fit differed substantially from out‐of‐sample model predictive performance; the best fitting models were rarely the best predictive models. Careful selection of predictor variables reduced the bias in parameter estimates, and there was no single best model across our three data sets. Testing and comparing multiple model forms is important. We developed an R package with a formula interface for straightforward fitting and validation of hierarchical, non‐linear growth models. Our intent is to encourage thorough testing of multiple growth model forms and an increased emphasis on assessing model fit relative to a model's purpose.     

Development and validation of experimental protocols for use of cardinal models for prediction of microorganism growth in food products

An experimental protocol to validate secondary-model application to foods was suggested. Escherichia coli, Listeria monocytogenes, Bacillus cereus, Clostridium perfringens, and Salmonella were observed in various food categories, such as meat, dairy, egg, or seafood products. The secondary model validated in this study was based on the gamma concept, in which the environmental factors temperature, pH, and water activity (aw) were introduced as individual terms with microbe-dependent parameters, and the effect of foodstuffs on the growth rates of these species was described with a food- and microbe-dependent parameter. This food-oriented approach was carried out by challenge testing, generally at 15 and 10 degrees C for L. monocytogenes, E. coli, B. cereus, and Salmonella and at 25 and 20 degrees C for C. perfringens. About 222 kinetics in foods were generated. The results were compared to simulations generated by existing software, such as PMP. The bias factor was also calculated. The methodology to obtain a food-dependent parameter (fitting step) and therefore to compare results given by models with new independent data (validation step) is discussed in regard to its food safety application. The proposed methods were used within the French national program of predictive microbiology, Sym'Previus, to include challenge test results in the database and to obtain predictive models designed for microbial growth in food products.     

Mathematical Models for the Effects of pH, Temperature, and Sodium Chloride on the Growth of Bacillus stearothermophilus in Salty Carrots

Estimating the shelf life and safety of any food product is an important part of food product development. Predictive food microbiology reduces the time and expense associated with conventional challenge and shelf life testing. The purpose of this study was to characterize and model germination, outgrowth, and lag (GOL) time and the exponential growth rate (EGR) of Bacillus stearothermophilus in salty carrot medium (SCM) as a function of pH, temperature, and NaCl concentration. B. stearothermophilus is a spore-forming thermophilic organism associated with flat sour spoilage of canned foods. A split-split plot design was used to measure the effects and interactions of pH (5.5 to 7.0), temperature (45 to 60(deg)C), and NaCl (0 to 1%) on the growth kinetics of B. stearothermophilus in SCM. A total of 96 experiments were analyzed, with individual curve parameters determined by using the Gompertz equation. Quadratic polynomial models for GOL time and EGR of B. stearothermophilus in terms of temperature, pH, and NaCl were generated by response surface analysis. The r(sup2) values for the GOL time and EGR models were 0.917 and 0.916, respectively. These models provide an estimate of bacterial growth in response to combinations of the variables studied within the specified ranges. The models were used to predict GOL times and EGRs for additional experimental conditions. The accuracy of these predictions validated the model's predictive ability in SCM.     

Chemometric analysis of in-line multi-wavelength fluorescence measurements obtained during cultivations with a lipase producing Aspergillus oryzae strain

The filamentous fungus, Aspergillus oryzae, was cultivated in batch and fed-batch cultivations in order to investigate the use of multi-wavelength fluorescence for monitoring course of events during filamentous fungi cultivations. The A. oryzae strain applied expressed a fungal lipase from Thermomyces lanuginosus. Spectra of multi-wavelength fluorescence were collected every 5 min with the BioView system (DELTA, Denmark) and both explorative and predictive models, correlating the fluorescence data with cell mass and lipase activity, were built. During the cultivations, A. oryzae displayed dispersed hyphal growth and under these conditions no fouling of the multi-wavelength fluorescence sensor was observed. The scores of a parallel factor analysis (PARAFAC) model, based on the fluorescence spectra, gave clear evidence of, for example, the on-set of the feeding phase. The predictive models, estimating the cell mass, showed correlations between 0.73 and 0.97 with root mean square error of cross validation (RMSECV) values between 1.48 and 0.77 g . kg(-1). A model estimating the lipase activity was also constructed for the fed-batch cultivations with a correlation of 0.93. The results presented here clearly show that multi-wavelength fluorescence is a useful tool for monitoring fed-batch cultivations of filamentous fungi.     

A New Secondary Model Developed for the Growth Rate of <Emphasis Type="Italic">Escherichia coli</Emphasis> O157:H7 in Broth

This study was attempted to develop a new exponential sum model to describe the effect of temperature on growth rate (GR) of Escherichia coli O157:H7 in broth. The growth rates of E. coli O157:H7 at different storage temperatures (4, 10, 15, 20, 25, 30, and 35°C) estimated by fitting with the modified Gompertz model were used to develop secondary models such as square root model, Ratkowsky model and exponential sum model. Measures of coefficient of determination (R ²), root mean square error (RMSE) and the sum of squares due to error (SSE) were employed to compare the performances of these three secondary models. Based on these criteria, the developed exponential sum model showed the better goodness-of-fit and performance.