Individual cell heterogeneity as variability source in population dynamics of microbial inactivation

3-D stochastic modeling approach in thermal inactivation: estimation of thermal survival kinetics of Escherichia coli O157:H7 in a hamburger after exposure to desiccation stress

Desiccation tolerance of pathogenic bacteria is one strategy for survival in harsh environments, which has been studied extensively. However, the subsequent survival behavior of desiccation-stressed bacterial pathogens has not been clarified in detail. Herein, we demonstrated that the effect of desiccation stress on the thermotolerance of Escherichia coli O157:H7 in ground beef was limited, and its thermotolerance did not increase. E. coli O157:H7 was inoculated into a ground beef hamburger after exposure to desiccation stress. We combined a bacterial inactivation model with a heat transfer model to predict the survival kinetics of desiccation-stressed E. coli O157:H7 in a hamburger. The survival models were developed using the Weibull model for two-dimensional pouched thin beef patties (ca. 1 mm), ignoring the temperature gradient in the sample, and a three-dimensional thick beef patty (ca. 10 mm), considering the temperature gradient in the sample. The two-dimensional (2-D) and three-dimensional (3-D) models were subjected to stochastic variations of the estimated Weibull parameters obtained from 1,000 replicated bootstrapping based on isothermal experimental observations as uncertainties. Furthermore, the 3-D model incorporated temperature gradients in the sample calculated using the finite element method. The accuracies of both models were validated via experimental observations under non-isothermal conditions using 100 predictive simulations. The root mean squared errors in the log survival ratio of the 2-D and 3-D models for 100 simulations were 0.25-0.53 and 0.32-2.08, respectively, regardless of the desiccation stress duration (24 or 72 h). The developed approach will be useful for setting appropriate process control measures and quantitatively assessing food safety levels.IMPORTANCEAcquisition of desiccation stress tolerance in bacterial pathogens might increase thermotolerance as well and increase the risk of foodborne illnesses. If a desiccation-stressed pathogen enters a kneaded food product via cross-contamination from a food-contact surface and/or utensils, proper estimation of the internal temperature changes in the kneaded food during thermal processing is indispensable for predicting the survival kinetics of desiccation-stressed bacterial cells. Various survival kinetics prediction models that consider the uncertainty or variability of pathogenic bacteria during thermal processing have been developed. Furthermore, heat transfer processes in solid food can be estimated using finite element method software. The present study demonstrated that combining a heat transfer model with a bacterial inactivation model can predict the survival kinetics of desiccation-stressed bacteria in a ground meat sample, corresponding to the temperature gradient in a solid sample during thermal processing. Combining both modeling procedures would enable the estimation of appropriate bacterial survival kinetics in solid food.

Advancements in Predictive Microbiology: Integrating New Technologies for Efficient Food Safety Models

Predictive microbiology is a rapidly evolving field that has gained significant interest over the years due to its diverse application in food safety. Predictive models are widely used in food microbiology to estimate the growth of microorganisms in food products. These models represent the dynamic interactions between intrinsic and extrinsic food factors as mathematical equations and then apply these data to predict shelf life, spoilage, and microbial risk assessment. Due to their ability to predict the microbial risk, these tools are also integrated into hazard analysis critical control point (HACCP) protocols. However, like most new technologies, several limitations have been linked to their use. Predictive models have been found incapable of modeling the intricate microbial interactions in food colonized by different bacteria populations under dynamic environmental conditions. To address this issue, researchers are integrating several new technologies into predictive models to improve efficiency and accuracy. Increasingly, newer technologies such as whole genome sequencing (WGS), metagenomics, artificial intelligence, and machine learning are being rapidly adopted into newer-generation models. This has facilitated the development of devices based on robotics, the Internet of Things, and time-temperature indicators that are being incorporated into food processing both domestically and industrially globally. This study reviewed current research on predictive models, limitations, challenges, and newer technologies being integrated into developing more efficient models. Machine learning algorithms commonly employed in predictive modeling are discussed with emphasis on their application in research and industry and their advantages over traditional models.

Is skipping the definition of primary and secondary models possible? Prediction of Escherichia coli O157 growth by machine learning

To predict bacterial population behavior in food, statistical models with specific function form have been applied in the field of predictive microbiology. Modelers need to consider the linear or non-linear relationship between the response and explanatory variables in the statistical modeling approach. In the present study, we focused on machine learning methods to skip definition of primary and secondary structure model. Support vector regression, extremely randomized trees regression, and Gaussian process regression were used to predict population growth of Escherichia coli O157 at 15 and 25 °C without defining the primary and secondary models. Furthermore, the support vector regression model was applied to predict small population of bacteria cells with probability theory. The model performance of the machine learning models were nearly equal to that of the current statistical models. Machine learning models have a potential for predicting bacterial population behavior.

The Inclusion of the Food Microstructural Influence in Predictive Microbiology: State-of-the-Art

Predictive microbiology has steadily evolved into one of the most important tools to assess and control the microbiological safety of food products. Predictive models were traditionally developed based on experiments in liquid laboratory media, meaning that food microstructural effects were not represented in these models. Since food microstructure is known to exert a significant effect on microbial growth and inactivation dynamics, the applicability of predictive models is limited if food microstructure is not taken into account. Over the last 10-20 years, researchers, therefore, developed a variety of models that do include certain food microstructural influences. This review provides an overview of the most notable microstructure-including models which were developed over the years, both for microbial growth and inactivation.

The COM-Poisson Process for Stochastic Modeling of Osmotic Inactivation Dynamics of Listeria monocytogenes

Controlling harmful microorganisms, such as Listeria monocytogenes, can require reliable inactivation steps, including those providing conditions (e.g., using high salt content) in which the pathogen could be progressively inactivated. Exposure to osmotic stress could result, however, in variation in the number of survivors, which needs to be carefully considered through appropriate dispersion measures for its impact on intervention practices. Variation in the experimental observations is due to uncertainty and biological variability in the microbial response. The Poisson distribution is suitable for modeling the variation of equi-dispersed count data when the naturally occurring randomness in bacterial numbers it is assumed. However, violation of equi-dispersion is quite often evident, leading to over-dispersion, i.e., non-randomness. This article proposes a statistical modeling approach for describing variation in osmotic inactivation of L. monocytogenes Scott A at different initial cell levels. The change of survivors over inactivation time was described as an exponential function in both the Poisson and in the Conway-Maxwell Poisson (COM-Poisson) processes, with the latter dealing with over-dispersion through a dispersion parameter. This parameter was modeled to describe the occurrence of non-randomness in the population distribution, even the one emerging with the osmotic treatment. The results revealed that the contribution of randomness to the total variance was dominant only on the lower-count survivors, while at higher counts the non-randomness contribution to the variance was shown to increase the total variance above the Poisson distribution. When the inactivation model was compared with random numbers generated in computer simulation, a good concordance between the experimental and the modeled data was obtained in the COM-Poisson process.

Bayesian Generalized Linear Model for Simulating Bacterial Inactivation/Growth Considering Variability and Uncertainty

Conventional regression analysis using the least-squares method has been applied to describe bacterial behavior logarithmically. However, only the normal distribution is used as the error distribution in the least-squares method, and the variability and uncertainty related to bacterial behavior are not considered. In this paper, we propose Bayesian statistical modeling based on a generalized linear model (GLM) that considers variability and uncertainty while fitting the model to colony count data. We investigated the inactivation kinetic data of Bacillus simplex with an initial cell count of 10⁵ and the growth kinetic data of Listeria monocytogenes with an initial cell count of 10⁴. The residual of the GLM was described using a Poisson distribution for the initial cell number and inactivation process and using a negative binomial distribution for the cell number variation during growth. The model parameters could be obtained considering the uncertainty by Bayesian inference. The Bayesian GLM successfully described the results of over 50 replications of bacterial inactivation with average of initial cell numbers of 10¹, 10², and 10³ and growth with average of initial cell numbers of 10^–1, 10⁰, and 10¹. The accuracy of the developed model revealed that more than 90% of the observed cell numbers except for growth with initial cell numbers of 10¹ were within the 95% prediction interval. In addition, parameter uncertainty could be expressed as an arbitrary probability distribution. The analysis procedures can be consistently applied to the simulation process through fitting. The Bayesian inference method based on the GLM clearly explains the variability and uncertainty in bacterial population behavior, which can serve as useful information for risk assessment related to food borne pathogens.

Effect οf οxygen availability and pH οn adaptive acid tolerance response of immobilized Listeria monocytogenes in structured growth media

The aim of the present study was to evaluate the effect of oxygen availability (aerobic, hypoxic and anoxic conditions) and sub-optimal pH (6.2 and 5.5) in a structured medium (10% w/V gelatin) on the growth of two immobilized L. monocytogenes strains (C5, 6179) at 10 °C and their subsequent acid resistance (pH 2.0, e.g., gastric acidity). Anaerobic conditions resulted in lower bacterial population (P < 0.05) (7.8-8.2 log CFU/mL) at the end of storage than aerobic and hypoxic environment (8.5-9.0 log CFU/mL), a phenomenon that was intensified at lower pH (5.5), where no significant growth was observed for anaerobically grown cultures. Prolonged habituation of L. monocytogenes (15 days) at both pH increased its acid tolerance resulting in max. 10 times higher t_4D (appx. 60 min). The combined effect though of oxygen availability and suboptimal pH on L. monocytogenes acid resistance was found to vary with the strain. Anoxically grown cultures at pH 5.5 exhibited the lowest tolerance towards lethal acid stress, with countable survivors occurring only until 20 min of exposure at pH 2.0. Elucidating the role of oxygen limiting conditions, often encountered in structured foods, on acid resistance of L. monocytogenes, would assist in assessing the capacity of L. monocytogenes originated from different food-related niches to withstand gastric acidity and possibly initiate infection.

Application of growth rate from kinetic model to calculate stochastic growth of a bacteria population at low contamination level

Traditional predictive microbiology is not suited for cell growth predictions for low-level contamination, where individual cell heterogeneity becomes apparent. Accordingly, we simulated a stochastic birth process of bacteria population using kinetic parameters. We predicted the variation in behavior of Salmonella enterica serovar Typhimurium cells at low inoculum density. The modeled cells were grown in tryptic soy broth at 25 °C. Kinetic growth parameters were first determined empirically for an initial cell number of 10⁴ cells. Monte Carlo simulation based on the growth kinetics and Poisson distribution for different initial cell numbers predicted the results of 50 replicate growth experiments with the initial cell number of 1, 10, and 64 cells. Indeed, measured behavior of 85% cells fell within the 95% prediction area of the simulation. The calculations link the kinetic and stochastic birth process with Poisson distribution. The developed model can be used to calculate the probability distribution of population size for exposure assessment and for the evaluation of a probability that a pathogen would exceed critical contamination level during food storage.

Assessing the survival and sublethal injury kinetics of Listeria monocytogenes under different food processing-related stresses

The foodborne pathogen L. monocytogenes can be present in food processing environments where it is exposed to various stressors. These antimicrobial factors, which aim to eliminate the pathogen, can induce sub-lethal injury to the bacterial cells. In the present study, we investigated the efficacy of different treatments (stresses) relevant to food processing and preservation as well as sanitation methods, in generating sub-lethal injury at 4 °C and 20 °C to two L. monocytogenes strains, ScottA and EGDe. Additionally, we evaluated the survival and extent of L. monocytogenes injury after exposure to commonly used disinfectants (peracetic acid and benzalkonium chloride), following habituation in nutrient-deprived, high-salinity medium. Each stress had a different impact on the survival and injury kinetics of L. monocytogenes. The highest injury levels were caused by peracetic acid which, at 4 °C, generated high populations of injured cells without loss of viability. Other injury-inducing stresses were lactic acid and heating. Long-term habituation in nutrient-limited and high salinity medium (4 °C) and subsequent exposure to disinfectants resulted in higher survival and injury in benzalkonium chloride and increased survival, yet with lower injury levels, in peracetic acid at 20 °C. Taken together, these results highlight the potential food safety risk emerging from the occurrence of injured cells by commonly used food processing methods. Consequently, in order to accurately assess the impact of an antimicrobial method, its potential of inducing sublethal injury needs to be considered along with lethality.

A microscopy-based approach for determining growth probability and lag time of individual bacterial cells

The development of relevant predictive models for single-cell lag time and growth probability near growth limits is of critical importance for predicting pathogen behavior in foods. The classical methods for data acquisition in this field are based on turbidity measurements of culture media in microplate wells inoculated with approximately one bacterial cell per well. Yet, these methods are labour intensive and would benefit from higher throughput. In this study, we developed a quantitative experimental method using automated microscopy to determine the single-cell growth probability and lag time. The developed method consists of the use of direct cell observation with phase-contrast microscopy equipped with a 100× objective and a high-resolution device camera. The method is not a time-lapse method but is based on the observation of high numbers of colonies for a given time. Automation of image acquisition and image analysis was used to reach a high throughput. The single-cell growth probabilities and lag times of four strains of Listeria monocytogenes were determined at 4 °C. The microscopic method was shown to be a promising method for the determination of individual lag times and growth probability at the single-cell level.

Dynamic modelling of Legionella pneumophila thermal inactivation in water

A predictive mathematical model describing the effect of temperature on the inactivation of Legionella pneumophila in water was developed. Thermal inactivation of L. pneumophila was monitored under isothermal conditions (51 - 61°C). A primary log-linear model was fitted to the inactivation data and the estimated D values ranged from 0.23 to 25.31 min for water temperatures from 61 to 51°C, respectively. The effect of temperature on L. pneumophila inactivation was described using a secondary model, and the model parameters z value and D_ref (D-value at 55°C) were estimated at 5.54°C and 3.47 min, respectively. The developed model was further validated under dynamic temperature conditions mimicking various conditions of water thermal disinfection in plumbing systems. The results indicated that the model can satisfactorily predict thermal inactivation of the pathogen at dynamic temperature environments and effectively translate water temperature profiles to cell number reduction. The application of the model in combination with effective temperature monitoring could provide the basis of an integrated preventive approach for the effective control of L. pneumophila in plumbing systems.

Microbiological and process variability using biological indicators of inactivation (BIIs) based on Bacillus cereus spores of food and fish-based animal by-products to evaluate microwave heating in a pilot plant

Microwave processing can be a valid alternative to conventional heating for different types of products. It enables a more efficient heat transfer in the food matrix, resulting in higher quality products. However, for many food products a uniform temperature distribution is not possible because of heterogeneities in their physical properties and non-uniformtiy in the electric field pattern. Hence, the effectiveness of microwave inactivation treatments is influenced by both intrinsic (differences between cells) and extrinsic variability (non-uniform temperature). Interpreting the results of the process and considering its impact on microbial inactivation is essential to ensure effective and efficient processing. In this work, we quantified the variability in microbial inactivation attained in a microwave pasteurization treatment with a tunnel configuration at pilot-plant scale. The configuration of the equipment makes it impossible to measure the product temperature during treatment. For that reason, variability in microbial counts was measured using Biological Inactivation Indicators (BIIs) based on spherical particles of alginate inoculated with spores of Bacillus spp. The stability of the BIIs and the uncertainty associated to them was assessed using preliminary experiments in a thermoresistometer. Then, they were introduced in the food product to analyse the microbial inactivation in different points of the products during the microwave treatment. Experiments were made in a vegetable soup and a fish-based animal by-product (F-BP). The results show that the variation in the microbial counts was higher than expected based on the biological variability estimated in the thermoresistometer and the uncertainty of the BIIs. This is due to heterogeneities in the temperature field (measured using a thermographic camera), which were higher in the F-BP than in the vegetable soup. Therefore, for the process studied, extrinsic variability was more relevant than intrinsic variability. The methodology presented in this work can be a valid method to evaluate pasteurization treatments of foods processed by heating, providing valuable information of the microbial inactivation achieved. It can contribute to design microwave processes for different types of products and for product optimization.

Variability in microbial inactivation: From deterministic Bigelow model to probability distribution of single cell inactivation times

Phenotypic heterogeneity seems to be an important component leading to biological individuality and is of great importance in the case of microbial inactivation. Bacterial cells are characterized by their own resistance to stresses. This inherent stochasticity is reflected in microbial survival curve which, in this context, can be considered as cumulative probability distribution of lethal events. The objective of the present study was to present an overview on the assessment and quantification of variability in microbial inactivation originating from single cells and discuss this heterogeneity in the context of predicting microbial behavior and Risk assessment studies. The detailed knowledge of the distribution of the single cells' inactivation times can be the basis for stochastic inactivation models which, in turn, may be employed in a risk - based food safety approach.

Describing the Individual Spore Variability and the Parameter Uncertainty in Bacterial Survival Kinetics Model by Using Second-Order Monte Carlo Simulation

The objective of this study was to separately describe the fitting uncertainty and the variability of individual cell in bacterial survival kinetics during isothermal and non-isothermal thermal processing. The model describing bacterial survival behavior and its uncertainties and variabilities during non-isothermal inactivation was developed from survival kinetic data for Bacillus simplex spores under fifteen isothermal conditions. The fitting uncertainties in the parameters used in the primary Weibull model was described by using the bootstrap method. The variability of individual cells in thermotolerance and the true randomness in the number of dead cells were described by using the Markov chain Monte Carlo (MCMC) method. A second-order Monte Carlo (2DMC) model was developed by combining both the uncertainties and variabilities. The 2DMC model was compared with reduction behavior under three non-isothermal profiles for model validation. The bacterial death estimations were validated using experimentally observed surviving bacterial count data. The fitting uncertainties in the primary Weibull model parameters, the individual thermotolerance heterogeneity, and the true randomness of inactivated spore counts were successfully described under all the iso-thermal conditions. Furthermore, the 2DMC model successfully described the variances in the surviving bacterial counts during thermal inactivation for all three non-isothermal profiles. As a template for risk-based process designs, the proposed 2DMC simulation approach, which considers both uncertainty and variability, can facilitate the selection of appropriate thermal processing conditions ensuring both food safety and quality.

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.

Multilevel modelling as a tool to include variability and uncertainty in quantitative microbiology and risk assessment. Thermal inactivation of Listeria monocytogenes as proof of concept

Variability is inherent in biology and also substantial for microbial populations. In the context of food safety risk assessment, it refers to differences in the response of different bacterial strains (between-strain variability) and different cells (within-strain variability) to the same condition (e.g. inactivation treatment). However, its quantification based on empirical observations and its incorporation in predictive models is a challenge for both experimental design and (statistical) analysis. In this article we propose the use of multilevel models to quantify (different levels of) variability and uncertainty and include them in the predictions. As proof of concept, we analyse the microbial inactivation of Listeria monocytogenes to thermal treatments including different levels of variability (between-strain and within-strain) and uncertainty. The relationship between the microbial count and time was expressed using a (non-linear) Weibullian model. Moreover, we defined stochastic hypotheses to describe the different types of variation at the level of the kinetic parameters, as well as in the observations (microbial counts). The model parameters (kinetic parameters and variances) are estimated using Bayesian statistics. The multilevel approach was compared against an analogous, single-level model. The multilevel methodology shrinks extreme parameter estimates towards the mean according to uncertainty, thus mitigating overfitting. In addition, this approach enables to easily incorporate different levels of variation (between-strain and/or within-strain variability and/or uncertainty) in the predictions. On the other hand, multilevel (Bayesian) models are more complex to define, implement, analyse and communicate than single-level models. Nevertheless, their ability to incorporate different sources of variability in predictions make them very suitable for Quantitative Microbial Risk Assessment.

Transforming kinetic model into a stochastic inactivation model: Statistical evaluation of stochastic inactivation of individual cells in a bacterial population

Kinetic models performing point estimation are effective in predicting the bacterial behavior. However, the large variation of bacterial behavior appearing in a small number of cells, i.e. equal or less than 10² cells, cannot be expressed by point estimation. We aimed to predict the variation of Escherichia coli O157:H7 behavior during inactivation in acidified tryptone soy broth (pH3.0) through Monte Carlo simulation and evaluated the accuracy of the developed model. Weibullian fitted parameters were estimated from the kinetic survival data of E. coli O157:H7 with an initial cell number of 10⁵. A Monte Carlo simulation (100 replication) based on the obtained Weibullian parameters and the Poisson distribution of initial cell numbers successfully predicted the results of 50 replications of bacterial inactivation with initial cell numbers of 10¹, 10², and 10³ cells. The accuracy of the simulation revealed that more than 83% of the observed survivors were within predicted range in all condition. 90% of the distribution in survivors with initial cells less than 100 is equivalent to a Poisson distribution. This calculation transforms the traditional microbial kinetic model into probabilistic model, which can handle bacteria number as discrete probability distribution. The probabilistic approach would utilize traditional kinetic model towards exposure assessment.

Modeling Salmonella Typhimurium Inactivation in Dry-Fermented Sausages: Previous Habituation in the Food Matrix Undermines UV-C Decontamination Efficacy

The effects of previous Salmonella Typhimurium habituation to an Italian-style salami concerning pathogen resistance against ultraviolet-C light (UV-C) treatment were modeled in order to establish treatment feasibility for the decontamination of dry-fermented sausage. S. Typhimurium following 24 h habituation in fermented sausage (habituated cells) or non-habituation (non-habituated cells) were exposed to increasing UV-C radiation treatment times. The Weibull model was the best fit for describing S. Typhimurium UV-C inactivation. Heterogeneity in UV-C treatment susceptibilities within the S. Typhimurium population was observed, revealing intrinsic persistence in a sub-population. UV-C radiation up to 1.50 J/cm2 was a feasible treatment for dry-fermented sausage decontamination, as the matrices retained instrumental color and lipid oxidation physiochemical characteristics. However, habituation in the sausage matrix led to a 14-fold increase in the UV-C dose required to achieve the first logarithm reduction (δ value) in S. Typhimurium population. The results indicate that, although UV-C radiation might be considered an efficient method for dry-fermented sausage decontamination, effective doses should be reconsidered in order to reach desirable food safety parameters while preserving matrix quality.

Predictive Modeling of Microbial Behavior in Food

Microorganisms can contaminate food, thus causing food spoilage and health risks when the food is consumed. Foods are not sterile; they have a natural flora and a transient flora reflecting their environment. To ensure food is safe, we must destroy these microorganisms or prevent their growth. Recurring hazards due to lapses in the handling, processing, and distribution of foods cannot be solved by obsolete methods and inadequate proposals. They require positive approach and resolution through the pooling of accumulated knowledge. As the industrial domain evolves rapidly and we are faced with pressures to continually improve both products and processes, a considerable competitive advantage can be gained by the introduction of predictive modeling in the food industry. Research and development capital concerns of the industry have been preserved by investigating the plethora of factors able to react on the final product. The presence of microorganisms in foods is critical for the quality of the food. However, microbial behavior is closely related to the properties of food itself such as water activity, pH, storage conditions, temperature, and relative humidity. The effect of these factors together contributing to permitting growth of microorganisms in foods can be predicted by mathematical modeling issued from quantitative studies on microbial populations. The use of predictive models permits us to evaluate shifts in microbial numbers in foods from harvesting to production, thus having a permanent and objective evaluation of the involving parameters. In this vein, predictive microbiology is the study of the microbial behavior in relation to certain environmental conditions, which assure food quality and safety. Microbial responses are evaluated through developed mathematical models, which must be validated for the specific case. As a result, predictive microbiology modeling is a useful tool to be applied for quantitative risk assessment. Herein, we review the predictive models that have been adapted for improvement of the food industry chain through a built virtual prototype of the final product or a process reflecting real-world conditions. It is then expected that predictive models are, nowadays, a useful and valuable tool in research as well as in industrial food conservation processes.

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.

Stochastic modeling of variability in survival behavior of Bacillus simplex spore population during isothermal inactivation at the single cell level using a Monte Carlo simulation

The control of bacterial reduction is important to maintain food safety during thermal processing. The goal of this study was to illustrate and describe variability in bacterial population behavior during thermal processing as a probability distribution based on individual cell heterogeneity regarding heat resistance. Toward this end, we performed a Monte Carlo simulation via computer, and compared and validated the simulated estimations with observed values. Weibullian fitted parameters were estimated from the kinetic survival data of Bacillus simplex during thermal treatment at 94 °C. The variability in reductions of bacterial sporular populations was illustrated using Monte Carlo simulation based on the Weibull distribution of the parameters. In particular, variabilities in viable spore counts and survival probability of the B. simplex spore population were simulated in various replicates. We also experimentally determined the changes in survival probability and distributions of survival spore counts; notably, these were successfully predicted by the Monte Carlo simulation based on the kinetic parameters. The kinetic parameter-based Monte Carlo simulation could thus successfully illustrate bacterial population behavior variability during thermal processing as a probability distribution. The simulation approach may contribute to improving food quality through risk-based processing designs and enhance risk assessment model accuracy.

Tail or artefact? Illustration of the impact that uncertainty of the serial dilution and cell enumeration methods has on microbial inactivation

The estimation of the concentration of microorganisms in a sample is crucial for food microbiology. For instance, it is essential for prevalence studies, challenge tests (growth and/or inactivation studies) or microbial risk assessment. The application of serial dilutions followed by viable counts in Petri dishes is probably the most extended experimental methodology for this purpose. However, this enumeration technique is also a source of uncertainty. In this article, the uncertainty of the serial dilution and viable count methodology related to the sampling error is analyzed, as well as the approximation of the microbial concentration by the number of colonies in a Petri dish. We analyze from a theoretical point of view (statistical analysis) the application of the binomial and Poisson models, demonstrating that the Poisson distribution increases the variance when used to model individual serial dilutions. On the other hand, the binomial model produces unbiased results. Therefore, the Poisson distribution is only applicable when it is a good approximation of the binomial distribution, so the use of the latter is recommended. The relevance of this uncertainty is demonstrated by Monte Carlo simulations of a generic microbial inactivation experiment, where the only source of uncertainty/variability considered is the one generated by serial plating and viable cell enumeration. Due to both the uncertainty of the methodology and the omission of zero-count plates because of the log-transformation, the simulated survival curve can have a tail. Therefore, this phenomenon, which is usually attributed to biological variability, can be to some extent an artefact of the experimental design and/or methodology.

Heterogeneity of single cell inactivation: Assessment of the individual cell time to death and implications in population behavior

A direct microscopic time-lapse method, using appropriate staining for cell viability in a confocal scanning laser microscope, was used for the direct assessment of Salmonella Agona individual cell inactivation in small two-dimensional colonies exposed to osmotic stress. Individual cell inactivation times were fitted to a variety of continuous distributions using @Risk software. The best fitted distribution (LogLogistic) was further used to predict the inactivation of Salmonella populations of various initial levels using Monte Carlo simulation. The simulation results showed that the variability in inactivation kinetics is negligible for concentrations down to 100 cells and the population behavior can be described with a deterministic model. As the concentration decreases below 100 cells, however, the variability increases significantly indicating that the traditional D-value used in deterministic first order kinetic models is not valid. At a second stage, single cell behavior was monitored in larger three dimensional colonies. The results showed that colony size can affect the inactivation pattern. The effect of colony size on microbial inactivation was confirmed with validation experiments which showed a higher inactivation rate for populations consisting of single cells or small colonies compared to those consisting of cells organized in larger colonies.

Stochastic simulation for death probability of bacterial population considering variability in individual cell inactivation time and initial number of cells

Decimal reduction time (D-value) based on the first-order survival kinetics model is not sufficient for reliable estimation of the bacterial survivors of inactivation treatment because the model does not consider inactivation curvature. However, even though doubt exists in the calculation of D-value, it is still widely used for risk assessment and sterilisation time estimation. This paper proposes an approach for estimating the time-to-inactivation and death probability of bacterial population that considers individual cell heterogeneity and initial number of cells via computer simulation. In the proposed approach, Weibull and Poisson distributions are respectively used to provide individual cell inactivation time variability and initial number of cells variability. Our simulation results show that the time-to-inactivation significantly depends on kinetics curvature and initial number of cells. For example, with increases in the initial number of cells, the respective variance of the time-to-inactivation of log-linear, concave downward curve, and concave upward curve remains constant, decreases, and increases, respectively. The death probability contour plot was successfully generated via our computer simulation approach without using D-value estimation. Further, the death probability calculated using our stochastic approach was virtually the same as that obtained using inactivation kinetics. We validated the simulation by using literature data for acid inactivation of Salmonella population. The results of this study indicate that inactivation curvature can replace D-value extrapolation to estimate the death probability of bacterial population. Further, our computer simulation facilitates realistic estimation of the time-to-inactivation of bacterial population. The R code used for the above stochastic calculation is outlined.

Relevance of the Induced Stress Resistance When Identifying the Critical Microorganism for Microbial Risk Assessment

Decisions regarding microbial risk assessment usually have to be carried out with incomplete information. This is due to the large number of possible scenarios and the lack of specific data for the problem considered. Consequently, risk assessment studies are based on the information obtained with a small number of bacterial cells which are considered the most heat resistant and/or more capable of multiplying during storage. The identification of the most resistant strains is usually based on D and z-values, normally estimated from isothermal experiments. This procedure omits the potential effect that the shape of the dynamic thermal profile applied in industry has on the microbial inactivation. One example of such effects is stress acclimation, which is related to a physiological response of the cells during sub-lethal treatments that increases their resistance. In this article, we use a recently published mathematical model to compare the development of thermal resistance for Escherichia coli K12 MG1655 and E. coli CECT 515 using inactivation data already published for these strains. Based only on the isothermal experiments, E. coli K12 MG1655 would be identified as more resistant to the thermal treatment than the CECT 515 strain in the 50-65°C temperature range. However, we conclude that stress acclimation is strain (and/or media)-dependent; the CECT 515 strain has a higher capacity for developing a stress acclimation than K12 MG1655 (300% increase of the D-value for CECT 515, 50% for K12 MG1655). It, thus, has the potential to be more resistant to the thermal treatment than the K12 MG1655 strain for some conditions allowing acclimation. A methodology is proposed to identify for which conditions this may be the case. After calibrating the model parameters representing acclimation using real experimental data, the applicability of the proposed approach is demonstrated using numerical simulations, showing how the CECT 515 strain can be more resistant for some heating profiles. Consequently, the most resistant bacterial strain to a dynamic heating profile should not be identified based only on isothermal experiments (D- and z-value). The relevance of stress acclimation for the treatment studied should also be evaluated.

Characterization of damage on Listeria innocua surviving to pulsed light: Effect on growth, DNA and proteome

The effect of pulsed light treatment on the lag phase and the maximum specific growth rate of Listeria innocua was determined in culture media at 7 °C. Fluences of 0.175, 0.350 and 0.525 J/cm² were tested. The lag phase of the survivors increased as fluence did, showing significant differences for all the doses; an 8.7-fold increase was observed at 0.525 J/cm². Pulsed light decreased the maximum specific growth rate by 38% at the same fluence. Both parameters were also determined by time-lapse microscopy at 25 °C in survivors to 0.525 J/cm², with an increase of 13-fold of the lag phase and a 45% decrease of the maximum specific growth rate. The higher the fluence, the higher the variability of both parameters was. To characterize pulsed light damage on L. innocua, the formation of dimers on DNA was assessed, and a proteomic study was undertaken. In cells treated with 0.525 J/cm², cyclobutane pyrimidine dimers and pyrimidine (6-4) pyrimidone photoproducts were detected at 5:1 ratio. Pulsed light induced the expression of three proteins, among them the general stress protein Ctc. Furthermore, treated cells showed an up-regulation of proteins related to metabolism of nucleotides and fatty acids, as well as with translation processes, whereas flagellin and some glucose metabolism proteins were down-regulated. Differences in the proteome of the survivors could contribute to explain the mechanisms of adaptation of L. innocua after pulsed light treatment.

Inactivation Kinetics of Pathogens during Thermal Processing in Acidified Broth and Tomato Purée (pH 4.5)

Thermal inactivation kinetics for single strains of Shiga toxin-producing Escherichia coli (STEC), Listeria monocytogenes, and Salmonella enterica were measured in acidified tryptic soy broth (TSB; pH 4.5) heated at 54°C. Inactivation curves also were measured for single-pathogen five-strain cocktails of E. coli O157:H7, L. monocytogenes, and S. enterica heated in tomato purée (pH 4.5) at 52, 54, 56, and 58°C. Inactivation curves were fit using log-linear and nonlinear (Weibull) models. The Weibull model yields the time for a 5-log reduction (t*) and a curve shape parameter (β). Decimal reduction times (D-values) and thermal resistance constants (z-values) from the two models were compared by defining t* = 5D* for the Weibull model. When the log-linear and Weibull models match at the 5-log reduction time, then t* = 5D* = 5D and D = D*. In 18 of 20 strains heated in acidified TSB, D and D* for the two models were not significantly different, although nonlinearity was observed in 35 of 60 trials. Similarly, in 51 of 52 trials for pathogen cocktails heated in tomato purée, D and D* were not significantly different, although nonlinearity was observed in 31% of trials. At a given temperature, D-values for S. enterica << L. monocytogenes < E. coli O157:H7 in tomato purée (pH 4.5). When using the two models, z-values calculated from the D-values were not significantly different for a given pathogen. Across all pathogens, z-values for E. coli O157:H7 and S. enterica were not different but were significantly lower than the z-values for L. monocytogenes. These results are useful for supporting process filings for tomato-based acidified food products with pH 4.5 and below and are relevant to small processors of tomato-based acidified canned foods who do not have the resources to conduct research on and validate pathogen lethality.

Modeling Stochastic Variability in the Numbers of Surviving Salmonella enterica, Enterohemorrhagic Escherichia coli, and Listeria monocytogenes Cells at the Single-Cell Level in a Desiccated Environment

Despite effective inactivation procedures, small numbers of bacterial cells may still remain in food samples. The risk that bacteria will survive these procedures has not been estimated precisely because deterministic models cannot be used to describe the uncertain behavior of bacterial populations. We used the Poisson distribution as a representative probability distribution to estimate the variability in bacterial numbers during the inactivation process. Strains of four serotypes of Salmonella enterica, three serotypes of enterohemorrhagic Escherichia coli, and one serotype of Listeria monocytogenes were evaluated for survival. We prepared bacterial cell numbers following a Poisson distribution (indicated by the parameter λ, which was equal to 2) and plated the cells in 96-well microplates, which were stored in a desiccated environment at 10% to 20% relative humidity and at 5, 15, and 25°C. The survival or death of the bacterial cells in each well was confirmed by adding tryptic soy broth as an enrichment culture. Changes in the Poisson distribution parameter during the inactivation process, which represent the variability in the numbers of surviving bacteria, were described by nonlinear regression with an exponential function based on a Weibull distribution. We also examined random changes in the number of surviving bacteria using a random number generator and computer simulations to determine whether the number of surviving bacteria followed a Poisson distribution during the bacterial death process by use of the Poisson process. For small initial cell numbers, more than 80% of the simulated distributions (λ = 2 or 10) followed a Poisson distribution. The results demonstrate that variability in the number of surviving bacteria can be described as a Poisson distribution by use of the model developed by use of the Poisson process.

IMPORTANCE We developed a model to enable the quantitative assessment of bacterial survivors of inactivation procedures because the presence of even one bacterium can cause foodborne disease. The results demonstrate that the variability in the numbers of surviving bacteria was described as a Poisson distribution by use of the model developed by use of the Poisson process. Description of the number of surviving bacteria as a probability distribution rather than as the point estimates used in a deterministic approach can provide a more realistic estimation of risk. The probability model should be useful for estimating the quantitative risk of bacterial survival during inactivation.

A Theoretical Analysis for Assessing the Variability of Secondary Model Thermal Inactivation Kinetic Parameters

Traditionally, for the determination of the kinetic parameters of thermal inactivation of a heat labile substance, an appropriate index is selected and its change is measured over time at a series of constant temperatures. The rate of this change is described through an appropriate primary model and a secondary model is applied to assess the impact of temperature. By this approach, the confidence intervals of the estimates of the rate constants are not taken into account. Consequently, the calculated variability of the secondary model parameters can be significantly lower than the actual variability. The aim of this study was to demonstrate the influence of the variability of the primary model parameters in establishing the confidence intervals of the secondary model parameters. Using a Monte Carlo technique and assuming normally distributed DT values (parameter associated with a primary inactivation model), the error propagating on the DTref and z-values (secondary model parameters) was assessed. When DT confidence intervals were broad, the secondary model's parameter variability was appreciably high and could not be adequately estimated through the traditional deterministic approach that does not take into account the variation on the DT values. In such cases, the proposed methodology was essential for realistic estimations.

Do bacterial cell numbers follow a theoretical Poisson distribution? Comparison of experimentally obtained numbers of single cells with random number generation via computer simulation

We investigated a bacterial sample preparation procedure for single-cell studies. In the present study, we examined whether single bacterial cells obtained via 10-fold dilution followed a theoretical Poisson distribution. Four serotypes of Salmonella enterica, three serotypes of enterohaemorrhagic Escherichia coli and one serotype of Listeria monocytogenes were used as sample bacteria. An inoculum of each serotype was prepared via a 10-fold dilution series to obtain bacterial cell counts with mean values of one or two. To determine whether the experimentally obtained bacterial cell counts follow a theoretical Poisson distribution, a likelihood ratio test between the experimentally obtained cell counts and Poisson distribution which parameter estimated by maximum likelihood estimation (MLE) was conducted. The bacterial cell counts of each serotype sufficiently followed a Poisson distribution. Furthermore, to examine the validity of the parameters of Poisson distribution from experimentally obtained bacterial cell counts, we compared these with the parameters of a Poisson distribution that were estimated using random number generation via computer simulation. The Poisson distribution parameters experimentally obtained from bacterial cell counts were within the range of the parameters estimated using a computer simulation. These results demonstrate that the bacterial cell counts of each serotype obtained via 10-fold dilution followed a Poisson distribution. The fact that the frequency of bacterial cell counts follows a Poisson distribution at low number would be applied to some single-cell studies with a few bacterial cells. In particular, the procedure presented in this study enables us to develop an inactivation model at the single-cell level that can estimate the variability of survival bacterial numbers during the bacterial death process.

Individual cell heterogeneity in Predictive Food Microbiology: Challenges in predicting a "noisy" world

Gene expression is a fundamentally noisy process giving rise to a significant cell to cell variability at the phenotype level. The phenotypic noise is manifested in a wide range of microbial traits. Heterogeneous behavior of individual cells is observed at the growth, survival and inactivation responses and should be taken into account in the context of Predictive Food Microbiology (PMF). Recent methodological advances can be employed for the study and modeling of single cell dynamics leading to a new generation of mechanistic models which can provide insight into the link between phenotype, gene-expression, protein and metabolic functional units at the single cell level. Such models however, need to deal with an enormous amount of interactions and processes that influence each other, forming an extremely complex system. In this review paper, we discuss the importance of noise and present the future challenges in predicting the "noisy" microbial responses in foods.

Antibacterial Effect of Copper on Microorganisms Isolated from Bovine Mastitis

The antimicrobial properties of copper have been recognized for several years; applying these properties to the prevention of diseases such as bovine mastitis is a new area of research. The aim of the present study was to evaluate in vitro the antimicrobial activity of copper on bacteria isolated from subclinical and clinical mastitis milk samples from two regions in Chile. A total of 327 microorganisms were recovered between March and September 2013, with different prevalence by sample origin (25 and 75% from the central and southern regions of Chile, respectively). In the central region, Escherichia coli and coagulase negative Staphylococci (CNS) were the most frequently detected in clinical mastitis cases (33%), while in the southern region S. uberis, S. aureus, and CNS were detected with frequencies of 22, 21, and 18%, respectively. Antibiotic susceptibility studies revealed that 34% of isolates were resistant to one or more antibiotics and the resistance profile was different between bacterial species and origins of isolation of the bacteria. The minimum inhibitory concentration of copper (MIC-Cu) was evaluated in all the isolates; results revealed that a concentration as low as 250 ppm copper was able to inhibit the great majority of microorganisms analyzed (65% of isolates). The remaining isolates showed a MIC-Cu between 375 and 700 ppm copper, and no growth was observed at 1000 ppm. A linear relationship was found between the logarithm of viable bacteria number and time of contact with copper. With the application of the same concentration of copper (250 ppm), CNS showed the highest tolerance to copper, followed by S. uberis and S. aureus; the least resistant was E. coli. Based on these in vitro results, copper preparations could represent a good alternative to dipping solutions, aimed at preventing the presence and multiplication of potentially pathogenic microorganisms involved in bovine mastitis disease.