Describing Uncertainty in Salmonella Thermal Inactivation Using Bayesian Statistical Modeling

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.

The Most Probable Curve method - A robust approach to estimate kinetic models from low plate count data resulting in reduced uncertainty

A novel method is proposed for fitting microbial inactivation models to data on liquid media: the Most Probable Curve (MPC) method. It is a multilevel model that makes a separation between the "true" microbial concentration according to the model, the "actual" concentration in the media considering chance, and the actual counts on the plate. It is based on the assumptions that stress resistance is homogeneous within a microbial population, and that there is no aggregation of microbial cells. Under these assumptions, the number of colonies in/on a plate follows a Poisson distribution with expected value depending on the proposed kinetic model, the number of dilutions and the plated volume. The novel method is compared against (non)linear regression based on a normal likelihood distribution (traditional method), Poisson regression and gamma-Poisson regression using data on the inactivation of Listeria monocytogenes. The conclusion is that the traditional method has limitations when the data includes plates with low (or zero) cell counts, which can be mitigated using more complex (discrete) likelihoods. However, Poisson regression uses an unrealistic likelihood function, making it unsuitable for survivor curves with several log-reductions. Gamma-Poisson regression uses a more realistic likelihood function, even though it is based mostly on empirical hypotheses. We conclude that the MPC method can be used reliably, especially when the data includes plates with low or zero counts. Furthermore, it generates a more realistic description of uncertainty, integrating the contribution of the plating error and reducing the uncertainty of the primary model parameters. Consequently, although it increases modelling complexity, the MPC method can be of great interest in predictive microbiology, especially in studies focused on variability analysis.

Competitive growth kinetics of Campylobacter jejuni, Escherichia coli O157:H7 and Listeria monocytogenes with enteric microflora in a small-intestine model

Aims

The biological events occurring during human digestion help to understand the mechanisms underlying the dose–response relationships of enteric bacterial pathogens. To better understand these events, we investigated the growth and reduction behaviour of bacterial pathogens in an in vitro model simulating the environment of the small intestine.

Methods and Results

The foodborne pathogens Campylobacter jejuni, Listeria monocytogenes and Escherichia coli O157:H7 were cultured with multiple competing enteric bacteria. Differences in the pathogen's growth kinetics due to the relative amount of competing enteric bacteria were investigated. These growth differences were described using a mathematical model based on Bayesian inference. When pathogenic and enteric bacteria were inoculated at 1 log CFU per ml and 9 log CFU per ml, respectively, L. monocytogenes was inactivated over time, while C. jejuni and E. coli O157:H7 survived without multiplying. However, as pathogen inocula were increased, its inhibition by enteric bacteria also decreased.

Conclusions

Although the growth of pathogenic species was inhibited by enteric bacteria, the pathogens still survived.

Significance and Impact of the Study

Competition experiments in a small‐intestine model have enhanced understanding of the infection risk in the intestine and provide insights for evaluating dose–response relationships.

Implementing a new dose-response model for estimating infection probability of Campylobacter jejuni based on the key events dose-response framework

Understanding the dose-response relationship between ingested pathogenic bacteria and infection probability is a key factor for appropriate risk assessment of foodborne pathogens. The objectives of this study were to develop and validate a novel mechanistic dose-response model for Campylobacter jejuni and simulate the underlying mechanism of foodborne illness during digestion. Bacterial behavior in the human gastrointestinal environment, including survival at low pH in the gastric environment after meals, transition to intestines, and invasion to intestinal tissues, was described using a Bayesian statistical model based on the reported experimental results of each process while considering physical food types (liquid or solid) and host age (young adult or elderly). Combining the models in each process, the relationship between pathogen intake and the infection probability of C. jejuni was estimated and compared with reported epidemiological dose-response relationships. Taking food types and host age into account, the prediction range of the infection probability of C. jejuni successfully covered the reported dose-response relationships from actual C. jejuni outbreaks. According to sensitivity analysis of predicted infection probabilities, the host age factor and the food type factor have relatively higher relevance than other factors. Thus, the developed Key Events Dose Response Framework can derive novel information for quantitative microbiological risk assessment in addition of dose-response relationship. The developed framework is potentially applicable to other pathogens to quantify the dose-response relationship from experimental data obtained from digestion. Importance Based on the mechanistic approach called Key Events Dose Response Framework alternative to previous non-mechanistic approach, the dose-response models for infection probability of C. jejuni were developed considering with age of people who take pathogen and food type. The developed predictive framework illustrated highly accurate prediction of dose (minimum difference 0.21 log CFU) for a certain infection probability compared with the previously reported dose-response relationship. In addition, the developed prediction procedure revealed that the dose-response relationship strongly depends on food type as well as host age. The implementation of Key Event Dose Response Framework will mechanistically and logically reveal the dose-response relationship and provide useful information with quantitative microbiological risk assessment of C. jejuni on foods.

Application of a Bayesian Statistical Framework for Planetary Protection as a Means of Verifying Low-Biomass, Zero-Inflated Test Data from Spacecraft

Planetary Protection is applicable for missions to biologically sensitive targets of interest in the solar system. For robotic missions landing on the Martian surface, Earth-based biological contamination must be reduced, controlled, and monitored to adhere to forward planetary protection requirements. To address the overall biological load limit and microbial density requirements per spacecraft each component is tracked based on its manufacturing pedigree and/or directly assessed using a direct sampling technique with either a swab or wipe. The tracking and reporting of requirements compliance has varied from mission to mission and reporting of numbers has consistently leaned towards the conservative worst-case scenario. With an increase in the number of missions and mission complexities, the need to establish a technically sound, statistical, and biological solution that provides a single point solution which addresses the distribution of spacecraft contamination becomes critical. Select components of the InSight mission, launched in 2018, have been used as a test case to evaluate the efficacy of applying Bayesian statistics to planetary protection data sets. Eight representative components covering the various bounding cases of high and low surface area, biological count, and sampling devices were analyzed as well as an assembly level case to evaluate the rollup of directly sampled and manufacturing pedigree components. A Bayesian approach was developed leveraging different priors from the zero-inflated data sets and compared to the heritage and existing NASA bioburden assessment approaches. In addition, several non-informative priors were evaluated for use in performing bioburden calculations. The results have demonstrated a viable framework to enable a Bayesian statistical approach to be further developed and utilized for planetary protection requirements assessment.

Evaluation of Strain Variability in Inactivation of Campylobacter jejuni in Simulated Gastric Fluid by Using Hierarchical Bayesian Modeling

This study was conducted to quantitatively evaluate the variability of stress resistance in different strains of Campylobacter jejuni and the uncertainty of such strain variability. We developed Bayesian statistical models with multilevel analysis to quantify variability within a strain, variability between different strains, and the uncertainty associated with these estimates. Furthermore, we measured the inactivation of 11 strains of C. jejuni in simulated gastric fluid with low pH, using the Weibullian survival model. The model was first developed for separate pH conditions and then analyzed over a range of pH levels. We found that the model parameters developed under separate pH conditions exhibited a clear dependence of survival on pH. In addition, the uncertainty of the variability between different strains could be described as the joint distribution of the model parameters. The latter model, including pH dependency, accurately predicted the number of surviving cells in individual as well as multiple strains. In conclusion, variabilities and uncertainties in inactivation could be simultaneously evaluated and interpreted via a probabilistic approach based on Bayesian theory. Such hierarchical Bayesian models could be useful for understanding individual-strain variability in quantitative microbial risk assessment. IMPORTANCE Since microbial strains vary in their growth and inactivation patterns in food materials, it is important to accurately predict these patterns for quantitative microbial risk assessment. However, most previous studies in this area have used highly resistant strains, which could lead to inaccurate predictions. Moreover, variability, including measurement errors and variability within a strain and between different strains, can contribute to predicted individual-level outcomes. Therefore, a multilevel framework is required to resolve these levels of variability and estimate their uncertainties. We developed a Bayesian predictive model for the survival of Campylobacter jejuni under simulated gastric conditions taking into account the variabilities and uncertainties. We demonstrated a high correspondence between predictions from the model and empirical measurements. The modeling procedure proposed in this study recommends a novel framework for predicting pathogen behavior, which can help improve quantitative microbial risk assessment during food production and distribution.

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.

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.

A stochastic approach for modelling the effects of temperature on the growth rate of Bacillus cereus sensu lato

A stochastic model that predicts the maximum specific growth rate (μ_max) of Bacillus cereus sensu lato as a function of temperature was developed. The model integrates the intra-species variability by incorporating distributions of cardinal parameters (T_min, T_opt, T_max) in the model. Growth rate data were generated for 22 strains, covering 5 major phylogenetic groups of B. cereus, and their cardinal temperatures identified. Published growth rate data were also incorporated in the model fitting, resulting in a set of 33 strains. Based on their cardinal temperatures, we identified clusters of Bacillus cereus strains that show similar response to temperature and these clusters were considered separately in the stochastic model. Interestingly, the μ_opt values for psychrotrophic strains were found to be significantly lower than those obtained for mesophilic strains. The model developed within this work takes into account some correlations existing between parameters (μ_opt, T_min, T_opt, T_max). In particular, the relationship highlighted between the b-slope of the Ratkowsky model and T_min (doi: https://doi.org/10.3389/fmicb.2017.01890) was adapted to the case of the popular Cardinal Temperature Model. This resulted in a reduced model in which μ_opt is replaced by a function of T_min, T_opt and 2 strain-independent parameters. A correlation between the T_min parameter and the experimental minimal growth temperature was also highlighted and integrated in the model for improved predictions near the temperature growth limits. Compared to the classical approach, the model developed in this study leads to improved predictions for temperatures around T_min and more realistic tails for the predicted distributions of μ_max. It can be useful for describing the variability of the Bacillus cereus Group in Quantitative Microbial Risk Assessment (QMRA). An example of application of the stochastic model to Reconstituted Infant Formulae (RIF) was proposed.

Modeling Invasion of Campylobacter jejuni into Human Small Intestinal Epithelial-Like Cells by Bayesian Inference

Current approaches used for dose-response modeling of low-dose exposures of pathogens rely on assumptions and extrapolations. These models are important for quantitative microbial risk assessment of food. A mechanistic framework has been advocated as an alternative approach for evaluating dose-response relationships. The objectives of this study were to investigate the invasion behavior of Campylobacter jejuni, which could arise as a foodborne illness even if there are low counts of pathogens, into Caco-2 cells as a model of intestinal cells and to develop a mathematical model for invading cell counts to reveal a part of the infection dose-response mechanism. Monolayer-cultured Caco-2 cells and various concentrations of C. jejuni in culture were cocultured for up to 12 h. The numbers of C. jejuni bacteria invading Caco-2 cells were determined after coculture for different time periods. There appeared to be a maximum limit to the invading bacterial counts, which showed an asymptotic exponential increase. The invading bacterial counts were higher with higher exposure concentrations (maximum, 5.0 log CFU/cm²) than with lower exposure concentrations (minimum, 0.6 log CFU/cm²). In contrast, the ratio of invading bacteria (number of invading bacteria divided by the total number of bacteria exposed) showed a similar trend regardless of the exposure concentration. Invasion of C. jejuni into intestinal cells was successfully demonstrated and described by the developed differential equation model with Bayesian inference. The model accuracy showed that the 99% prediction band covered more than 97% of the observed values. These findings provide important information on mechanistic pathogen dose-response relationships and an alternative approach for dose-response modeling.IMPORTANCE One of the infection processes of C. jejuni, the invasion behavior of the bacteria in intestinal epithelial cells, was revealed, and a mathematical model for prediction of the cell-invading pathogen counts was developed for the purpose of providing part of a dose-response model for C. jejuni based on the infection mechanism. The developed predictive model showed a high accuracy of more than 97% and successfully described the C. jejuni invading counts. The bacterial invasion predictive model of this study will be essential for the development of a dose-response model for C. jejuni based on the infection mechanism.

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.