Comparing uncertainty resulting from two-step and global regression procedures applied to microbial growth models

The Use of Predictive Microbiology for the Prediction of the Shelf Life of Food Products

Microbial shelf life refers to the duration of time during which a food product remains safe for consumption in terms of its microbiological quality. Predictive microbiology is a field of science that focuses on using mathematical models and computational techniques to predict the growth, survival, and behaviour of microorganisms in food and other environments. This approach allows researchers, food producers, and regulatory bodies to assess the potential risks associated with microbial contamination and spoilage, enabling informed decisions to be made regarding food safety, quality, and shelf life. Two-step and one-step modelling approaches are modelling techniques with primary and secondary models being used, while the machine learning approach does not require using primary and secondary models for describing the quantitative behaviour of microorganisms, leading to the spoilage of food products. This comprehensive review delves into the various modelling techniques that have found applications in predictive food microbiology for estimating the shelf life of food products. By examining the strengths, limitations, and implications of the different approaches, this review provides an invaluable resource for researchers and practitioners seeking to enhance the accuracy and reliability of microbial shelf life predictions. Ultimately, a deeper understanding of these techniques promises to advance the domain of predictive food microbiology, fostering improved food safety practices, reduced waste, and heightened consumer confidence.

Comparison of modelling approaches for the prediction of kinetic growth parameters of Pseudomonas spp. in oyster mushroom (Pleurotus ostreatus)

In predictive microbiology, primary and secondary models can be used to predict microbial growth, usually in a two-step modelling approach. The inverse dynamic modelling approach is an alternative method to direct modelling methods, in which the primary and secondary models are fitted simultaneously from non-isothermal data, minimising experimental effort and costs. Thus, the main aim of the present study was to compare the prediction capabilities of the mathematical modelling approaches used for calculating growth kinetics of microorganisms in predictive food microbiology field. For this purpose, the bacterial growth data of Pseudomonas spp. in oyster mushroom (Pleurotus ostreatus) subjected to isothermal and non-isothermal storage temperatures were collected from previously published growth curves. Temperature-dependent kinetic growth parameters (maximum specific growth rate 'µmax' and lag phase duration 'λ') were described as a function of storage temperature using the direct two-step, direct one-step and inverse dynamic modelling approach based on Baranyi and Huang models. The fitting capability of the modelling approaches was separately compared, and the one-step modelling approach for the direct methods provided better goodness of fit results regardless of used primary models, which leads the Huang model with being RMSE = 0.226 and R2adj = 0.949 became best for direct methods. Like seen in direct methods, the Huang model gave better goodness of fit results than Baranyi model for inverse method. Results revealed there was no significant difference (p > 0.05) between the growth kinetic parameters obtained from direct one-step modelling approach and inverse modelling approaches based on the Huang model. Satisfactorily statistical indexes show that the inverse dynamic modelling approach can be reliably used as an alternative way of describing the growth behaviour of Pseudomonas spp. in oyster mushroom in a fast and minimum labour effort.

Prediction of Pseudomonas spp. Population in Food Products and Culture Media Using Machine Learning-Based Regression Methods

Machine learning approaches are alternative modelling techniques to traditional modelling equations used in predictive food microbiology and utilise algorithms to analyse large datasets that contain information about microbial growth or survival in various food matrices. These approaches leverage the power of algorithms to extract insights from the data and make predictions regarding the behaviour of microorganisms in different food environments. The objective of this study was to apply various machine learning-based regression methods, including support vector regression (SVR), Gaussian process regression (GPR), decision tree regression (DTR), and random forest regression (RFR), to estimate bacterial populations. In order to achieve this, a total of 5618 data points for Pseudomonas spp. present in food products (beef, pork, and poultry) and culture media were gathered from the ComBase database. The machine learning algorithms were applied to predict the growth or survival behaviour of Pseudomonas spp. in food products and culture media by considering predictor variables such as temperature, salt concentration, water activity, and acidity. The suitability of the algorithms was assessed using statistical measures such as coefficient of determination (R²), root mean square error (RMSE), bias factor (Bf), and accuracy (A_f). Each of the regression algorithms showed appropriate estimation capabilities with R² ranging from 0.886 to 0.913, RMSE from 0.724 to 0.899, B_f from 1.012 to 1.020, and A_f from 1.086 to 1.101 for each food product and culture medium. Since the predictive capability of RFR was the best among the algorithms, externally collected data from the literature were used for RFR. The external validation process showed statistical indices of B_f ranging from 0.951 to 1.040 and A_f ranging from 1.091 to 1.130, indicating that RFR can be used for predicting the survival and growth of microorganisms in food products. Therefore, machine learning approaches can be considered as an alternative to conventional modelling methods in predictive microbiology. However, it is important to highlight that the prediction power of the machine learning regression method directly depends on the dataset size, and it requires a large dataset to be employed for modelling. Therefore, the modelling work of this study can only be used for the prediction of Pseudomonas spp. in specific food products (beef, pork, and poultry) and culture medium with certain conditions where a large dataset is available.

Modelling growth of Bacillus cereus in paneer by one-step parameter estimation

Bacillus cereus phylogenetic group III and IV strains are commonly associated with food products and cause toxin mediated foodborne diseases. These pathogenic strains have been identified from milk and dairy products, such as reconstituted infant formula and several cheeses. Paneer is a fresh, soft cheese originating from India that is prone to foodborne pathogen contamination, such as by Bacillus cereus. However, there are no reported studies of B. cereus toxin formation in paneer or predictive models quantifying growth of the pathogen in paneer under different environmental conditions. This study assessed enterotoxin-producing potential of B. cereus group III and IV strains, isolated from dairy farm environments, in fresh paneer. Growth of a four-strain cocktail of toxin-producing B. cereus strains was measured in freshly prepared paneer incubated at 5-55 °C and modelled using a one-step parameter estimation combined with bootstrap re-sampling to generate confidence intervals for model parameters. The pathogen grew in paneer between 10 and 50 °C and the developed model fit the observed data well (R² = 0.972, RMSE = 0.321 log₁₀ CFU/g). The cardinal parameters for B. cereus growth in paneer along with the 95% confidence intervals were: μ_opt 0.812 log₁₀ CFU/g/h (0.742, 0.917); T_opt is 44.177 °C (43.16, 45.49); T_min is 4.405 °C (3.973, 4.829); T_max is 50.676 °C (50.367, 51.144). The model developed can be used in food safety management plans and risk assessments to improve safety of paneer while also adding to limited information on B. cereus growth kinetics in dairy products.

Growth and Non-Thermal Inactivation of Staphylococcus aureus in Sliced Dry-Cured Ham in Relation to Water Activity, Packaging Type and Storage Temperature

Dry-cured ham (DCH) could support the growth of Staphylococcus aureus as a halotolerant bacterium, which may compromise the shelf-stability of the product according to the growth/no growth boundary models and the physicochemical parameters of commercial DCH. In the present study, the behavior of S. aureus is evaluated in sliced DCH with different water activity (a_w 0.861-0.925), packaged under air, vacuum, or modified atmosphere (MAP), and stored at different temperatures (2-25 °C) for up to 1 year. The Logistic and the Weibull models were fitted to data to estimate the primary kinetic parameters for the pathogen Log₁₀ increase and Log₁₀ reduction, respectively. Then, polynomial models were developed as secondary models following their integration into the primary Weibull model to obtain a global model for each packaging. Growth was observed for samples with the highest a_w stored at 20 and 25 °C in air-packaged DCH. For lower a_w, progressive inactivation of S. aureus was observed, being faster at the lowest temperature (15 °C) for air-packaged DCH. In contrast, for vacuum and MAP-packaged DCH, a higher storage temperature resulted in faster inactivation without a significant effect of the product a_w. The results of this study clearly indicate that the behavior of S. aureus is highly dependent on factors such as storage temperature, packaging conditions and product a_w. The developed models provide a management tool for evaluating the risk associated with DCH and for preventing the development of S. aureus by selecting the most appropriate packaging according to a_w range and storage temperature.

Development of a New Modelling Approach and Performance Evaluation of Meta-heuristic Optimization Algorithms for the Prediction of Kinetic Growth Parameters for Pseudomonas spp. in Fish

The main aim of the current work was to build up a new mathematical modelling approach in predictive food microbiology field for the prediction of growth kinetics of microorganisms. For this purpose, the bacterial growth data of Pseudomonas spp. in whole fish (gilt-head seabream) subjected to isothermal and non-isothermal storage temperatures were collected from previously published growth curves. Maximum specific growth rate (1/h) and lag phase duration (h) were described as a function of storage temperature using the direct two-step, direct one-step and inverse dynamic modelling approaches based on various meta-heuristic optimization algorithms. The fitting capability of the modelling approaches and employed optimization algorithms was separately compared, and the one-step modelling approach for the direct methods and the Bayesian optimization method for the used algorithms provided the best goodness of fit results. These two were then further processed in validation step. The inverse dynamic modelling approach based on the Bayesian optimization algorithm yielded satisfactorily statistical indexes (1.02 > Bias factor > 1.09 and 1.07 > Accuracy factor > 1.13), which indicates it can be reliably used as an alternative way of describing the growth behaviour of Pseudomonas spp. in fish in a fast and efficient manner with minimum labour effort.

Development and validation of a one-step modelling approach for the determination of chicken meat shelf-life based on the growth kinetics of Pseudomonas spp

The main objective of the present study was to investigate the effect of storage temperature on aerobically stored chicken meat spoilage using the two-step and one-step modelling approaches involving different primary models namely the modified Gompertz, logistic, Baranyi and Huang models. For this purpose, growth data points of Pseudomonas spp. were collected from published studies conducted in aerobically stored chicken meat product. Temperature-dependent kinetic parameters (maximum specific growth rate 'µ_max' and lag phase duration 'λ') were described as a function of storage temperature through the Ratkowsky model based on the different primary models. Then, the fitting capability of both modelling approaches was compared taking into account root mean square error, adjusted coefficient of determination (adjusted-R²) and corrected Akaike information criterion. The one-step modelling approach showed considerably improved fitting capability regardless of the used primary model. Finally, models developed from the one-step modelling approach were validated for the maximum growth rate data extracted from independent published literature using the statistical indexes Bias (B_f) and Accuracy (A_f) factors. The best prediction capability was obtained for the Baranyi model with B_f and A_f being very close to 1. The shelf-life of chicken meat as a function of storage temperature was predicted using both modelling approaches for the Baranyi model.

Effect of Alternative Preservation Steps and Storage on Vitamin C Stability in Fruit and Vegetable Products: Critical Review and Kinetic Modelling Approaches

Vitamin C, a water-soluble compound, is a natural antioxidant in many plant-based products, possessing important nutritional benefits for human health. During fruit and vegetable processing, this bioactive compound is prone to various modes of degradation, with temperature and oxygen being recognised as the main factors responsible for this nutritional loss. Consequently, Vitamin C is frequently used as an index of the overall quality deterioration of such products during processing and post-processing storage and handling. Traditional preservation methods, such as thermal processing, drying and freezing, are often linked to a substantial Vitamin C loss. As an alternative, novel techniques or a combination of various preservation steps ("hurdles") have been extensively investigated in the recent literature aiming at maximising Vitamin C retention throughout the whole product lifecycle, from farm to fork. In such an integrated approach, it is important to separately study the effect of each preservation step and mathematically describe the impact of the prevailing factors on Vitamin C stability, so as to be able to optimise the processing/storage phase. In this context, alternative mathematical approaches have been applied, including more sophisticated ones that incorporate parameter uncertainties, with the ultimate goal of providing more realistic predictions.

Risk management tool to define a corrective storage to enhance Salmonella inactivation in dry fermented sausages

The resistance of Salmonella to the harsh conditions occurring in shelf-stable dry fermented sausages (DFS) poses a food safety challenge for producers. The present study aimed to model the behaviour of Salmonella in acid (with starter culture) and low-acid (without starter culture) DFS as a function of a_w and storage temperature in order to build a decision supporting tool supporting the design of a corrective storage strategy to enhance the safety of DFS. Salmonella spp. were inoculated in the raw meat batter at ca. 6 Log cfu/g with a cocktail of 3 strains (CTC1003, CTC1022 and CTC1754) just before mixing with the other ingredients and additives. After stuffing, sausages were fermented and ripened following industrial processing conditions. Different drying-times were applied to obtain three batches with different a_w (0.88, 0.90 and 0.93). Afterwards, DFS were stored at 4, 8, 15 and 25 °C for a maximum of three months and Salmonella spp. were periodically enumerated. The Weibull model was fitted to Log counts data to estimate inactivation kinetic parameters. The impact of temperature and a_w on the primary inactivation parameters was evaluated using a polynomial equation. The results of the challenge tests showed that Salmonella spp. levels decreased during storage at all the assayed conditions, from 0.8 Log (in low-acid DFS at 4 °C) up to 6.5 Log (in acid DFS at 25 °C). The effect of both a_w and temperature was statistically significant. Delta (δ) parameter decreased by decreasing a_w and increasing temperature, while the shape (p) parameter ranged from above 1 (concave) at 10 °C to below 1 at 25 °C (convex). A common secondary model for the p parameter was obtained for each type of DFS, acid and low-acid, indicating that acidification during the production of DFS affected the time for the first Log reduction (δ) during the subsequent storage, but not the overall shape (p parameter) of the inactivation. The developed models covered representative of real conditions, such as Salmonella contamination in the raw materials and its adaptation to the harsh processing conditions. The good predictive performance shown when applying the models to independent data (i.e. up to 80% of the predictions within the 'Acceptable Simulation Zone' for acid sausages) makes them a suitable and reliable risk management tool to support manufacturers to assess and design a lethality treatment (i.e. corrective storage) to enhance the Salmonella inactivation in the product before DFS are released to the market.

Holistic Approach to the Uncertainty in Shelf Life Prediction of Frozen Foods at Dynamic Cold Chain Conditions

Systematic kinetic modeling is required to predict frozen systems behavior in cold dynamic conditions. A one-step procedure, where all data are used simultaneously in a non-linear algorithm, is implemented to estimate the kinetic parameters of both primary and secondary models. Compared to the traditional two-step methodology, more precise estimates are obtained, and the calculated parameter uncertainty can be introduced in realistic shelf life predictions, as a tool for cold chain optimization. Additionally, significant variability of the real distribution/storage conditions is recorded, and must be also incorporated in a kinetic prediction scheme. The applicability of the approach is theoretically demonstrated in an analysis of data on frozen green peas Vitamin C content, for the calculation of joint confidence intervals of kinetic parameters. A stochastic algorithm is implemented, through a double Monte Carlo scheme incorporating the temperature variability during distribution, drawn from cold chain databases. Assuming a distribution scenario of 130 days in the cold chain, 93 ± 110 days remaining shelf life was predicted compared to 180 days assumed based on the use by date. Overall, through the theoretical case study investigated, the uncertainty of models’ parameters and cold chain dynamics were incorporated into shelf life assessment, leading to more realistic predictions.

One-Step Analysis for Listeria monocytogenes Growth in Ready-to-Eat Braised Beef at Dynamic and Static Conditions

This study aimed to estimate the growth parameters of Listeria monocytogenes growth in ready-to-eat (RTE) braised beef by one-step dynamic and static kinetic analysis. The Baranyi model and cardinal parameters model were integrated into a dynamic and static model to estimate the kinetic parameters under one dynamic condition (-20 to 40.0°C) and eight static conditions (4, 8, 15, 20, 30, 35, 37, and 40°C). Based on the dynamic and static methods, the respective dynamic and static results for estimated growth boundaries of L. monocytogenes in RTE braised beef were from -2.5 and -2.7°C to 40.5 and 40.7°C with optimal specific growth rates of 1.078 and 0.913 per h at temperatures of 35.7 and 35.0°C. Temperature effects on the specific growth rate and lag period were developed and used to simulate the change of the physiological state of inocula during the bacterial growth. Subsequently, three additional dynamic temperature profiles were implemented for external validation. The root mean square error of the model developed by dynamic regression (0.19 log CFU/g) is slightly better than that of the model developed by static regression (0.23 log CFU/g). Comparing the validation results, one-step dynamic analysis might be a preferable method for prediction, especially when the growth approaches the stationary phase. Generally, both one-step dynamic and static analyses could be used to accurately predict L. monocytogenes growth in RTE braised beef under fluctuating temperatures.

Describing Uncertainty in Salmonella Thermal Inactivation Using Bayesian Statistical Modeling

Uncertainty analysis is the process of identifying limitations in scientific knowledge and evaluating their implications for scientific conclusions. In the context of microbial risk assessment, the uncertainty in the predicted microbial behavior can be an important component of the overall uncertainty. Conventional deterministic modeling approaches which provide point estimates of the pathogen's levels cannot quantify the uncertainty around the predictions. The objective of this study was to use Bayesian statistical modeling for describing uncertainty in predicted microbial thermal inactivation of Salmonella enterica Typhimurium DT104. A set of thermal inactivation data in broth with water activity adjusted to 0.75 at 9 different temperature conditions obtained from the ComBase database (www.combase.cc) was used. A log-linear microbial inactivation was used as a primary model while for secondary modeling, a linear relation between the logarithm of inactivation rate and temperature was assumed. For comparison, data were fitted with a two-step and a global Bayesian regression. Posterior distributions of model's parameters were used to predict Salmonella thermal inactivation. The combination of the joint posterior distributions of model's parameters allowed the prediction of cell density over time, total reduction time and inactivation rate as probability distributions at different time and temperature conditions. For example, for the time required to eliminate a Salmonella population of about 107 CFU/ml at 65°C, the model predicted a time distribution with a median of 0.40 min and 5th and 95th percentiles of 0.24 and 0.60 min, respectively. The validation of the model showed that it can describe successfully uncertainty in predicted thermal inactivation with most observed data being within the 95% prediction intervals of the model. The global regression approach resulted in less uncertain predictions compared to the two-step regression. The developed model could be used to quantify uncertainty in thermal inactivation in risk-based processing design as well as in risk assessment studies.

Cross-Laboratory Comparative Study of the Impact of Experimental and Regression Methodologies on Salmonella Thermal Inactivation Parameters in Ground Beef

Isothermal inactivation studies are commonly used to quantify thermal inactivation kinetics of bacteria. Meta-analyses and comparisons utilizing results from multiple sources have revealed large variations in reported thermal resistance parameters for Salmonella, even when in similar food materials. Different laboratory or regression methodologies likely are the source of methodology-specific artifacts influencing the estimated parameters; however, such effects have not been quantified. The objective of this study was to evaluate the effects of laboratory and regression methodologies on thermal inactivation data generation, interpretation, modeling, and inherent error, based on data generated in two independent laboratories. The overall experimental design consisted of a cross-laboratory comparison using two independent laboratories (Michigan State University and U.S. Department of Agriculture, Agricultural Research Service, Eastern Regional Research Center [ERRC] laboratories), both conducting isothermal Salmonella inactivation studies (55, 60, 62°C) in ground beef, and each using two methodologies reported in prior studies. Two primary models (log-linear and Weibull) with one secondary model (Bigelow) were fitted to the resultant data using three regression methodologies (two two-step regressions and a one-step regression). Results indicated that laboratory methodology impacted the estimated D60°C- and z-values (α = 0.05), with the ERRC methodology yielding parameter estimates ∼25% larger than the Michigan State University methodology, regardless of the laboratory. Regression methodology also impacted the model and parameter error estimates. Two-step regressions yielded root mean square error values on average 40% larger than the one-step regressions. The Akaike Information Criterion indicated the Weibull as the more correct model in most cases; however, caution should be used to confirm model robustness in application to real-world data. Overall, the results suggested that laboratory and regression methodologies have a large influence on resultant data and the subsequent estimation of thermal resistance parameters.

Analysing and modelling the growth behaviour of Listeria monocytogenes on RTE cooked meat products after a high pressure treatment at 400 MPa

Various predictive models are available for high pressure inactivation of Listeria monocytogenes in food, but currently available models do not consider the growth kinetics of surviving cells during the subsequent storage of products. Therefore, we characterised the growth of L. monocytogenes in sliced cooked meat products after a pressurization treatment. Two inoculum levels (10(7) or 10(4) CFU/g) and two physiological states before pressurization (freeze-stressed or cold-adapted) were evaluated. Samples of cooked ham and mortadella were inoculated, high pressure processed (400 MPa, 5 min) and subsequently stored at 4, 8 and 12 °C. The Logistic model with delay was used to estimate lag phase (λ) and maximum specific growth rate (μmax) values from the obtained growth curves. The effect of storage temperature on μmax and λ was modelled using the Ratkowsky square root model and the relative lag time (RLT) concept. Compared with cold-adapted cells the freeze-stressed cells were more pressure-resistant and showed a much longer lag phase during growth after the pressure treatment. Interestingly, for high-pressure inactivation and subsequent growth, the time to achieve a concentration of L. monocytogenes 100-fold (2-log) higher than the cell concentration prior to the pressure treatment was similar for the two studied physiological states of the inoculum. Two secondary models were necessary to describe the different growth behaviour of L. monocytogenes on ready-to-eat cooked ham (lean product) and mortadella (fatty product). This supported the need of a product-oriented approach to assess growth after high pressure processing. The performance of the developed predictive models for the growth of L. monocytogenes in high-pressure processed cooked ham and mortadella was evaluated by comparison with available data from the literature and by using the Acceptable Simulation Zone approach. Overall, 91% of the relative errors fell into the Acceptable Simulation Zone.

Effect of pH, water activity and gel micro-structure, including oxygen profiles and rheological characterization, on the growth kinetics of Salmonella Typhimurium

In this study, the growth of Salmonella Typhimurium in Tryptic Soy Broth was examined at different pH (4.50-5.50), water activity a(w) (0.970-0.992) and gelatin concentration (0%, 1% and 5% ) at 20 degrees C. Experiments in TSB with 0% gelatin were carried out in shaken erlenmeyers, in the weak 1% gelatin media in petri plates and in the firm 5% gelatin media in gel cassettes. A quantification of gel strength was performed by rheological measurements and the influence of oxygen supply on the growth of S. Typhimurium was investigated. pH, as well as a(w) as well as gelatin concentration had an influence on the growth rate. Both in broth and in gelatinized media, lowering pH or water activity caused a decrease of growth rate. In media with 1% gelatin a reduction of growth rate and maximal cell density was observed compared to broth at all conditions. However, the effects of decreasing pH and a(w) were less pronounced. A further increase in gelatin concentration to 5% gelatin caused a small or no additional drop of growth rate. The final oxygen concentration dropped from 5.5 ppm in stirred broth to anoxic values in petri plates, also when 0% and 5% gelatin media were tested in this recipient. Probably, not stirring the medium, which leads to anoxic conditions, has a more pronounced effect on the growth rate of S. Typhimurium then medium solidness. Finally, growth data were fitted with the primary model of Baranyi and Roberts [Baranyi, J. and Roberts, T. A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277-294]. An additional factor was introduced into the secondary model of Ross et al. [Ross, T. and Ratkowsky, D. A. and Mellefont, L. A. and McMeekin, T. A., 2003. Modelling the effects of temperature, water activity, pH and lactic acid concentration on the growth rate of Escherichia coli. International Journal of Food Microbiology 82, 33-43.] to incorporate the effect of gelatin concentration, next to the effect of pH and a(w). A two step and a global regression procedure were applied. Both procedures were able to fit the data well, but the global regression procedure led to smaller standard errors on the parameters.