Stochastic and Deterministic Model of Microbial Heat Inactivation

The probability of bacterial spores surviving a thermal process: The 12D myth and other issues with its quantitative assessment

The concepts of "D-value," "thermal death time" and "commercial sterility," innovative and useful at their inception, are based on untenable assumptions, notably that the log-linear isothermal inactivation model has universal applicability, that extrapolation over several orders of magnitude below the detection level is permissible, and that total microbial inactivation is theoretically impossible. Almost all commonly observed inactivation patterns, the log-linear is just a special case, can be described by both deterministic and fully stochastic models, examples of which are given. Unlike the deterministic, the stochastic models predict either complete elimination of the targeted cells or spores in realistic finite time, or residual survival. In most cases, the published survival data do not contain enough information to establish which actually happens. The microbial safety of thermally processed foods can be compromised not only by under-processing but also by a variety of mishaps whose occurrence probabilities are unrelated to the inactivation kinetics. Moreover, the available sampling plans to detect microbial contamination in sterilized containers through incubation alone are insensitive to levels of potential safety concerns. In principle, many of these issues could be resolved by developing new dramatically improved detection methods and/or verifiable methods to predict very low levels of microbial survival.

Deterministic and stochastic survival models of injured ozonated Giardia cysts

Giardia cysts exposed to short sublethal ozonation in lake waters continue to die-off well after the ozone complete dissipation. This delayed inactivation can be the manifestation of injured cysts' mortality, which the traditional Chick-Watson-Hom type models of disinfection do not account for. But it can be described by a slightly modified version of a general microbial survival model adapted for injured cysts or other targeted microorganisms surviving disinfection. The downward concavity of the cysts' semi-logarithmic survival ratio vs. time relationships suggests that the cysts' deaths had unimodal temporal distribution. Indeed, the cumulative (CDF) forms of the Weibull and lognormal distribution functions both had excellent fit to the experimental survival data. Such a survival pattern can also be described by a fully probabilistic model devised from the injured cysts' Markov chain, where the mortality's probability rate rises linearly with time. The stochastic model explains the ubiquitous observation that microbial survival curves become increasingly irregular and irreproducible as the number of survivors dwindles, regardless of their concavity degree and direction. Although based on ozonated Giardia cyst data, the concept should be applicable to the delayed mortality of other microorganisms surviving sublethal treatments of other kinds but unable to recover and/or multiply. KEY POINTS: • Deterministic and stochastic survival models can describe delayed inactivation. • The Weibull and lognormal distributions can describe cysts' times to mortality. • Stochastic model explains the progressively growing scatter in survival curves.

Microbial Dose-Response Curves and Disinfection Efficacy Models Revisited

The same term “dose-response curve” describes the relationship between the number of ingested microbes or their logarithm, and the probability of acute illness or death (type I), and between a disinfectant’s dose and the targeted microbe’s survival ratio (type II), akin to survival curves in thermal and non-thermal inactivation kinetics. The most common model of type I curves is the cumulative form of the beta-Poisson distribution which is sometimes indistinguishable from the lognormal or Weibull distribution. The most notable survival kinetics models in static disinfection are of the Chick-Watson-Hom’s kind. Their published dynamic versions, however, should be viewed with caution. A microbe population’s type II dose-response curve, static and dynamic, can be viewed as expressing an underlying spectrum of individual vulnerabilities (or resistances) to the particular disinfectant. Therefore, such a curve can be described mathematically by the flexible Weibull distribution, whose scale parameter is a function of the disinfectant’s intensity, temperature, and other factors. But where the survival ratio’s drop is so steep that the static dose-response curve resembles a step function, the Fermi distribution function becomes a suitable substitute. The utility of the CT (or Ct) concept primarily used in water disinfection is challenged on theoretical grounds and its limitations highlighted. It is suggested that stochastic models of microbial inactivation could be used to link the fates of individual viruses or bacteria to their manifestation in the survival curve’s shape. Although the emphasis is on viruses and bacteria, most of the discussion is relevant to fungi, protozoa, and perhaps worms too.

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.

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.

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.

Bridging the divide: a model-data approach to Polar and Alpine microbiology

Advances in microbial ecology in the cryosphere continue to be driven by empirical approaches including field sampling and laboratory-based analyses. Although mathematical models are commonly used to investigate the physical dynamics of Polar and Alpine regions, they are rarely applied in microbial studies. Yet integrating modelling approaches with ongoing observational and laboratory-based work is ideally suited to Polar and Alpine microbial ecosystems given their harsh environmental and biogeochemical characteristics, simple trophic structures, distinct seasonality, often difficult accessibility, geographical expansiveness and susceptibility to accelerated climate changes. In this opinion paper, we explain how mathematical modelling ideally complements field and laboratory-based analyses. We thus argue that mathematical modelling is a powerful tool for the investigation of these extreme environments and that fully integrated, interdisciplinary model-data approaches could help the Polar and Alpine microbiology community address some of the great research challenges of the 21st century (e.g. assessing global significance and response to climate change). However, a better integration of field and laboratory work with model design and calibration/validation, as well as a stronger focus on quantitative information is required to advance models that can be used to make predictions and upscale processes and fluxes beyond what can be captured by observations alone.

Modeling of fungal and bacterial spore germination under static and dynamic conditions

Isothermal germination curves, sigmoid and nonsigmoid, can be described by a variety of models reminiscent of growth models. Two of these, which are consistent with the percent of germinated spores being initially zero, were selected: one, Weibullian (or "stretched exponential"), for more or less symmetric curves, and the other, introduced by Dantigny's group, for asymmetric curves (P. Dantigny, S. P.-M. Nanguy, D. Judet-Correia, and M. Bensoussan, Int. J. Food Microbiol. 146:176-181, 2011). These static models were converted into differential rate models to simulate dynamic germination patterns, which passed a test for consistency. In principle, these and similar models, if validated experimentally, could be used to predict dynamic germination from isothermal data. The procedures to generate both isothermal and dynamic germination curves have been automated and posted as freeware on the Internet in the form of interactive Wolfram demonstrations. A fully stochastic model of individual and small groups of spores, developed in parallel, shows that when the germination probability is constant from the start, the germination curve is nonsigmoid. It becomes sigmoid if the probability monotonically rises from zero. If the probability rate function rises and then falls, the germination reaches an asymptotic level determined by the peak's location and height. As the number of individual spores rises, the germination curve of their assemblies becomes smoother. It also becomes more deterministic and can be described by the empirical phenomenological models.

Evaluation of a stochastic inactivation model for heat-activated spores of Bacillus spp

Heat activates the dormant spores of certain Bacillus spp., which is reflected in the "activation shoulder" in their survival curves. At the same time, heat also inactivates the already active and just activated spores, as well as those still dormant. A stochastic model based on progressively changing probabilities of activation and inactivation can describe this phenomenon. The model is presented in a fully probabilistic discrete form for individual and small groups of spores and as a semicontinuous deterministic model for large spore populations. The same underlying algorithm applies to both isothermal and dynamic heat treatments. Its construction does not require the assumption of the activation and inactivation kinetics or knowledge of their biophysical and biochemical mechanisms. A simplified version of the semicontinuous model was used to simulate survival curves with the activation shoulder that are reminiscent of experimental curves reported in the literature. The model is not intended to replace current models to predict dynamic inactivation but only to offer a conceptual alternative to their interpretation. Nevertheless, by linking the survival curve's shape to probabilities of events at the individual spore level, the model explains, and can be used to simulate, the irregular activation and survival patterns of individual and small groups of spores, which might be involved in food poisoning and spoilage.