Biomedical subjects

J F Van Impe

Publications and source records attributed to J F Van Impe.

At least 19 recordsLinked to original sources

PubMed2005

Reflections on the use of robust and least-squares non-linear regression to model challenge tests conducted in/on food products.

In this research, we question the straight-forward use of the classical sum of squared error criterion for identifying the typical parameters of a primary model (like growth rate mumax and lag time lambda) when applied to growth curves obtained in and on food products. Firstly, we base our reflections on 62 Listeria monocytogenes laboratory challenge tests collected in various environments (broth, crushed cold-smoked salmon, and surface of cold-smoked salmon slices). Whereas growth data in broth resulted in residual values consistent with a Gaussian distribution, growth data in the crushed product and even more on the surface of slices appeared different. Secondly, we propose the use of an alternative so-called robust non-linear regression method suitable when experimental error is non-normally distributed, which seems, according to this research, typical for microbial challenge tests in/on food products, and which lead to apparent outliers or leverage points in the experimental data. Properties of the robust regression procedure are illustrated on simulated data first, whereafter its use on the considered challenge tests is illustrated. To conclude, reflections on the assumptions and related realism underlying challenge tests and recommendations for fitting growth curves obtained in and on food products are presented.

Animals↗

PubMed2005

Predictive modelling and validation of Listeria innocua growth at superatmospheric oxygen and carbon dioxide concentrations.

The effect of superatmospheric oxygen and carbon dioxide concentrations on the growth of Listeria innocua, which was used as a model organism for the pathogen Listeria monocytogenes, was evaluated. The bacteria were grown on a nutrient agar surface at 7 degrees C. Three carbon dioxide levels (0%, 12.5% and 25%) were combined with different levels of high oxygen concentrations (above 20%) based on a mixture design. The applied oxygen concentrations did not significantly influence the growth. High CO2 concentrations, on the contrary, reduced the maximum specific growth rate and prolonged the lag time. An overall model to describe the growth of L. innocua under high carbon dioxide conditions was constructed based on nine growth experiments, using a weighted one-step regression procedure. The influence of carbon dioxide on lag time and maximum specific growth rate was described using Ratkowsky-type models and inserted in the Baranyi equation. The model described the growth very well. To assess the validity of the model, 14 additional experiments were carried out. There was a good correlation of the model predictions and observed validation data.

Carbon Dioxide↗

PubMed2005

Predictive modelling of the individual and combined effect of water activity and temperature on the radial growth of Fusarium verticilliodes and F. proliferatum on corn.

The major objective of this study was to develop validated models to describe the effect of a(w) and temperature on the radial growth on corn of the two major fumonisin producing Fusaria, namely Fusarium verticilliodes and F. proliferatum. The growth of these two isolates on corn was therefore studied at water activities between 0.810-0.985 and temperatures between 15 and 30 degrees C. Minimum a(w) for growth was 0.869 and 0.854 for F. verticilliodes and F. proliferatum, respectively. No growth took place at a(w) values equal to 0.831 and 0.838 for F. verticilliodes and F. proliferatum, respectively. The colony growth rates, g (mm d(-1)) were determined by fitting a flexible growth model describing the change in colony diameter (mm) with respect to time (days). Secondary models, relating the colony growth rate with a(w) or a(w) and temperature were developed. A third order polynomial equation and the linear Arrhenius-Davey model were used to describe the combined effect of temperature and a(w) on g. The combined modelling approaches, predicting g (mm d(-1)) at any a(w) and/or temperature were validated on independently collected data. All models proved to be good predictors of the growth rates of both isolates on maize within the experimental conditions. The third order polynomial equation had bias factors of 1.042 and 1.054 and accuracy factors of 1.128 and 1.380 for F. verticilliodes and F. proliferatum, respectively. The linear Arrhenius-Davey model had bias factors of 0.978 and 1.002 and accuracy factors of 1.098 and 1.122 for F. verticilliodes and F. proliferatum, respectively. The results confirm the general finding that a(w) has a greater influence on fungal growth than temperature. The developed models can be applied for the prevention of Fusarium growth on maize and the development of models that incorporate other factors important to mould growth on maize.

Adsorption↗

PubMed2005

Predicting the onset of filamentous bulking in biological wastewater treatment systems by exploiting image analysis information.

The performance of the activated sludge process is limited by the ability of the sedimentation tank (1) to separate the activated sludge from the treated effluent and (2) to concentrate it. Apart from bad operating strategies or poorly designed clarifiers, settling failures can mainly be attributed to filamentous bulking. Image analysis is a promising technique that can be used for early detection of filamentous bulking. The aim of this paper is therefore twofold. Foremost, correlations are sought between image analysis information (i.e., the total filament length per image, the mean form factor, the mean equivalent floc diameter, the mean floc roundness and the mean floc reduced radius of gyration) and classical measurements (i.e., the Sludge Volume Index (SVI)). Secondly, this information is both explored and exploited in order to identify dynamic ARX and state space-type models. Their performance is compared based on two criteria.

Algorithms↗

PubMed2005

GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves.

This contribution focuses on the presentation of GInaFiT (Geeraerd and Van Impe Inactivation Model Fitting Tool), a freeware Add-in for Microsoft Excel aiming at bridging the gap between people developing predictive modelling approaches and end-users in the food industry not familiar with or not disposing over advanced non-linear regression analysis tools. More precisely, the tool is useful for testing nine different types of microbial survival models on user-specific experimental data relating the evolution of the microbial population with time. As such, the authors believe to cover all known survivor curve shapes for vegetative bacterial cells. The nine model types are: (i) classical log-linear curves, (ii) curves displaying a so-called shoulder before a log-linear decrease is apparent, (iii) curves displaying a so-called tail after a log-linear decrease, (iv) survival curves displaying both shoulder and tailing behaviour, (v) concave curves, (vi) convex curves, (vii) convex/concave curves followed by tailing, (viii) biphasic inactivation kinetics, and (ix) biphasic inactivation kinetics preceded by a shoulder. Next to the obtained parameter values, the following statistical measures are automatically reported: standard errors of the parameter values, the Sum of Squared Errors, the Mean Sum of Squared Errors and its Root, the R(2) and the adjusted R(2). The tool can help the end-user to communicate the performance of food preservation processes in terms of the number of log cycles of reduction rather than the classical D-value and is downloadable via the KULeuven/BioTeC-homepage at the topic "Downloads" (Version 1.4, Release date April 2005).

Bacteria↗

PubMed2005

Cell division theory and individual-based modeling of microbial lag: part I. The theory of cell division.

This series of two papers deals with the theory of cell division and its implementation in an individual-based modeling framework. In this first part, the theory of cell division is studied on an individual-based level in order to learn more about the mechanistic principles behind microbial lag phenomena. While some important literature on cell division theory dates from 30 to 40 years ago, until now it has hardly been introduced in the field of predictive microbiology. Yet, it provides a large amount of information on how cells likely respond to changing environmental conditions. On the basis of this theory, a general theory on microbial lag behavior caused by a combination of medium and/or temperature changes has been developed in this paper. The proposed theory then forms the basis for a critical evaluation of existing modeling concepts for microbial lag in predictive microbiology. First of all, a more thorough definition can be formulated to define the lag time lambda and the previously only vaguely defined physiological state of the cells in terms of mechanistically defined parameters like cell mass, RNA or protein content, specific growth rate and time to perform DNA replication and cell division. On the other hand, existing predictive models are evaluated with respect to the newly developed theory. For the model of , a certain fitting parameter can also be related to physically meaningful parameters while for the model of [Augustin, J.-C., Rosso, L., Carlier, V.A. 2000. A model describing the effect of temperature history on lag time for Listeria monocytogenes. Int. J. Food Microbiol. 57, 169-181] a new, mechanistically based, model structure is proposed. A restriction of the proposed theory is that it is only valid for situations where biomass growth responds instantly to an environment change. The authors are aware of the fact that this assumption is not generally acceptable. Lag in biomass can be caused, for example, by a delayed synthesis of some essential growth factor (e.g., enzymes). In the second part of this series of papers [Dens, E.J., Bernaerts, K., Standaert, A.R., Kreft, J.-U., Van Impe, J.F., this issue. Cell division theory and individual-based modeling of microbial lag: part II. Modeling lag phenomena induced by temperature shifts. Int. J. Food Microbiol], the theory of cell division is implemented in an individual-based simulation program and extended to account for lags in biomass growth. In conclusion, the cell division theory applied to microbial populations in dynamic medium and/or temperature conditions provides a useful framework to analyze microbial lag behavior.

Algorithms↗

PubMed2005

Cell division theory and individual-based modeling of microbial lag: part II. Modeling lag phenomena induced by temperature shifts.

This paper is the second in a series of two, and studies microbial lag in cell number and/or biomass measurements caused by temperature changes with an individual-based modeling approach. For this purpose, the theory of cell division, as discussed in the first part of this series of research papers, was implemented in the individual-based modeling framework BacSim. Simulations of this model are compared with experimental data of Escherichia coli, growing in an aerated, glucose-rich medium and subjected to sudden temperature shifts. The premise of a constant cell volume under changing temperature conditions predicts no lag in cell numbers after the shift, in contrast to the experimental observations. Based on literature research, two biological mechanisms that could be responsible for the observed lag phenomena are proposed. The first assumes that the average cell volume depends on temperature while the second assumes that a lag in biomass growth occurs after the temperature shift. For a lag in cell number caused by an increased average cell volume, the cell biomass always increases at the maximal rate. Therefore, cells are evidently not stressed and do not have to adapt to the new conditions, as opposed to a lag in biomass growth. Implementation and simulation of both mechanisms are found to describe the experimental observations equally well. Therefore, further research is needed to distinguish between the two mechanisms. This can be done by observing, in addition to cell numbers, a measure for the average cell volumes. In conclusion, the individual-based modeling approach is a good methodology to investigate and test biological theories and assumptions. Also, based on the simulations, suggestions for further experimental observations can be made.

Algorithms↗

PubMed2005

Optimal experiment design for cardinal values estimation: guidelines for data collection.

Optimal experiment design for parameter estimation (OED/PE) is an interesting technique for modelling practices when aiming for maximum parameter estimation accuracy. Nowadays, experimental designs for secondary modelling within the field of predictive microbiology are mostly arbitrary or based on factorial design. The latter type of design is common practice in response surface modelling approaches. A number of levels of the factor(s) under study are selected and all possible treatment combinations are performed. It is however not always clear which levels and treatment combinations are most relevant. An answer to this question can be obtained from optimal experiment design for-in this particular case-parameter estimation. This technique is based on the extremisation of a scalar function of the Fisher information matrix. The type of scalar function determines the final focus of the optimised design. In this paper, optimal experiment designs are computed for the cardinal temperature model with inflection point (CTMI) and the cardinal pH model (CPM). A model output sensitivity analysis (depicting the sensitivity of the model output to a small change in the model parameters) yields a first indication of relevant temperature or pH treatments. Performed designs are: D-optimal design aiming for a maximum global parameter estimation accuracy (by minimising the determinant of the Fisher information matrix), and E-optimal design improving the confidence in the most uncertain model parameter (by maximising the smallest eigenvalue of the Fisher information matrix). Although lowering the information content of a set of experiments, boundary values on the design region need to be imposed during optimisation to exclude unworkable experiments and partly account for incorrect nominal parameter values. Opposed to the frequently applied equidistant or arbitrary treatment placement, optimal design results show that typically four informative temperature or pH levels are selected and replicate experiments are to be performed at these points. Informative experiments are typically placed at points with an extreme model output sensitivity.

Bacteria↗

PubMed2005

Quantifying microbial lag phenomena due to a sudden rise in temperature: a systematic macroscopic study.

The microbial lag phase is a complex and yet not completely understood phenomenon. Many studies on the microbial lag phase have been published but few report a systematic study; moreover, previous lag studies have involved the effect of multiple confounded factors. Here, the effect of sudden temperature rises on an exponentially growing Escherichia coli culture is systematically investigated. Experiments are performed in a computer-controlled bioreactor where E. coli K12 MG1655 is grown under aerobic conditions in Brain Heart Infusion (BHI) broth. This experimental set-up is used to characterise the effect of (i) the amplitude of the temperature shift, (ii) the pre-shift temperature level and (iii) the post-shift temperature level on the occurrence and length of a lag phase. Besides temperature, no other environmental changes take place at the moment of the temperature shift. To quantify the length of the induced lag phase, the experimental data are described with a common growth model. Depending on the three factors tested, a lag phase of more or less 1 h is induced or not. This lag/no lag behaviour can largely be explained by the existence of a normal physiological temperature range but also the amplitude of the temperature rise plays a role. It can be concluded that for the microorganism under study the lower boundary of the normal range lies approximately between 22.78 and 23.86 degrees C. It is shown that this boundary is no cut-off point, but rather a transition zone. Even more, repeated experiments at this boundary have yielded different results (lag or no lag). This observation points out that the mechanism triggering this lag phase is not absolute but may be subject to biological variability.

Adaptation, Physiological↗

PubMed2005

Environmental factors influencing the relationship between optical density and cell count for Listeria monocytogenes.

AIMS: The effect of temperature (2-30 degrees C), pH (4.8-7.4) and water activity (0.946-0.995) on the relationship between optical density (OD) at 600 nm and the plate count (CFU ml(-1)) was investigated for Listeria monocytogenes. METHODS AND RESULTS: Calibration curves, relating OD with plate counts, were collected by measuring the OD of consecutive one-half dilution series, before determining the cell density by classic plate count methods. The calibration curves were observed to be shifting in a parallel way, with increasing stress levels. Especially pH influenced the curve in a great extent, while the other variables were showing more synergetic effects. The reason for the shift was investigated by a microscopic viability test, showing a viability decrease with increasing stress levels, causing the shift of the calibration curve. In a last step a model was made describing the effect of environmental factors on the calibration curve, with different data transformations being tested. A polynomial equation was fitted to the data, taking into account a set of constraints to incorporate microbiological knowledge in the black box model. Hence, illogical interpolation results and overfitting of the data could be avoided. CONCLUSIONS: Different stress factors are affecting the relationship between the OD and the cell count of L. monocytogenes by lowering the cell viability. These effects could be modelled using a constrained polynomial model. SIGNIFICANCE AND IMPACT OF THE STUDY: The observed phenomena are important when calculating growth parameters, like growth rate and lag phase, based on OD data.

Calibration↗

PubMed2005

Mathematical modelling of small wastewater treatment plants: power and limitations.

Although mathematical modelling of biological wastewater treatment processes has proved to be valuable for large-scale WWTPs (wastewater treatment plants) little experience has been acquired in the mathematical modelling of small wastewater treatment plants. This paper seeks to evaluate the applicability of mathematical modelling on small systems, which are characterized by high fluctuations in organic and hydraulic loads and little possibility for control. In order to achieve this, the paper examines the different steps in a general modelling protocol. One important bottleneck for the general use of mathematical modelling of small systems that emerges is the frequent sampling and many analyses needed for characterization of the flows while its applicability is limited. On the other hand, the determination of the model structure of a small WWTP can be quite valuable. Experiments show that tracer tests should include tests with a highly varying influent flow rate to spot independent small internal flows as these can have a significant impact on the behaviour of peak concentrations throughout the system. In addition, the model structure determination can provide useful information on dead zones, short-circuiting and mixing behaviour in the plant.

Models, Theoretical↗

PubMed2004

Obtaining single cells: analysis and evaluation of an experimental protocol by means of a simulation model.

The research presented in this paper analyses a newly developed experimental protocol for isolating single cells by constructing a simulation model of the process. The protocol involves sequential 50% dilutions of a cell suspension in a microtiter plate, so that eventually, wells are obtained containing exactly one cell. The aim of this modelling study is (i) to gain insight in the governing mechanisms of the dilution process, (ii) to confirm experimental findings and (iii) to enable the prediction of an average outcome for future experiments. The model construction process is presented chronologically. The initial basic model simulates the experiment as a sequence of binomial processes, using Monte Carlo techniques. Statistical analysis of the results shows that aggregational factors need to be taken into account in the form of a lognormal distribution. Several issues involved in this adaptation are discussed. To fully account for cell aggregation in the dilution process, a cell clumping algorithm is built into the simulation model. Simulation data from the resulting model show similar statistical characteristics as the experimental data and yield reliable prediction intervals for the available experimental data. The simulation model is a useful tool to support experimental findings and predict the outcome of future experiments. Even more importantly, this study emphasises the importance of careful statistical analysis in single cell research. The impact of stochastic effects is considerably amplified at the low cell concentrations involved and needs to be taken into account in any modelling effort.

Bacteria↗

PubMed2004

Towards a novel class of predictive microbial growth models.

Food safety and quality are influenced by the presence (and possible proliferation) of pathogenic and spoilage microorganisms during the life cycle of the product (i.e., from the raw ingredients at the start of the production process until the moment of consumption). In order to simulate and predict microbial evolution in foods, mathematical models are developed in the field of predictive microbiology. In general, microbial growth is a self-limiting process, principally due to either (i) the exhaustion of one of the essential nutrients, and/or (ii) the accumulation of toxic products that inhibit growth. Nowadays, most mathematical models used in predictive microbiology do not explicitly incorporate this basic microbial knowledge. In this paper, a novel class of microbial growth models is proposed. In contrast with the currently used logistic type models, e.g., the model of Baranyi and Roberts [Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277-294], the novel model class explicitly incorporates nutrient exhaustion and/or metabolic waste product effects. As such, this novel model prototype constitutes an elementary building block to be extended in a natural way towards, e.g., microbial interactions in co-cultures (mediated by metabolic products) and microbial growth in structured foods (influenced by, e.g., local substrate concentrations). While under certain conditions the mathematical equivalence with classical logistic type models is clear and results in equal fitting capacities and parameter estimation quality (see Poschet et al. [Poschet, F., Vereecken, K.M., Geeraerd, A.H., Nicolai, B.M., Van Impe, J.F., 2004. Analysis of a novel class of predictive microbial growth models and application to co-culture growth. International Journal of Food Microbiology, this issue] for a more elaborated analysis in this respect), the biological interpretability and extendability represent the main added value.

Bacteria↗

PubMed2004

Analysis of a novel class of predictive microbial growth models and application to coculture growth.

In this paper, a novel class of microbial growth models is analysed. In contrast with the currently used logistic type models (e.g., the model of Baranyi and Roberts [Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277-294]), the novel model class, presented in Van Impe et al. (Van Impe, J.F., Poschet, F., Geeraerd, A.H., Vereecken, K.M., 2004. Towards a novel class of predictive microbial growth models. International Journal of Food Microbiology, this issue), explicitly incorporates nutrient exhaustion and/or metabolic waste product effects inducing stationary phase behaviour. As such, these novel model types can be extended in a natural way towards microbial interactions in cocultures and microbial growth in structured foods. Two illustrative case studies of the novel model types are thoroughly analysed and compared to the widely used model of Baranyi and Roberts. In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition and is described as a function of the pH-evolution. In the second case study, substrate exhaustion is the sole cause of the stationary phase. Finally, a more complex case study of a so-called P-model is presented, dealing with a coculture inhibition of Listeria innocua mediated by lactic acid production of Lactococcus lactis.

Coculture Techniques↗

PubMed2004

Estimating the probability and level of contamination with Salmonella of feed for finishing pigs produced in Switzerland--the impact of the production pathway.

Contaminated feed is a source of infection with Salmonella for livestock, including pigs. Because pigs rarely show clinical signs of salmonellosis, undetected carriers can enter the food production chain. In a "Farm to Fork" food safety concept, safe feed is the first step for ensuring safe food. Heat treatment or adding organic acids are process steps for reducing or eliminating a contamination with Salmonella. The aims of this study were (I) to estimate the probability and the level of Salmonella contamination in batches of feed for finishing pigs in Swiss mills and (II) to assess the efficacy of specific process steps for reducing the level of contamination with Salmonella. A quantitative release assessment was performed by gathering and combining data on the various parameters having an influence on the final contamination of feed. Fixed values and probability distributions attributed to these parameters were used as input values for a Monte Carlo simulation. The simulation showed that-depending on the production pathway-the probability that a batch of feed for finishing pigs contains Salmonella ranged from 34% (for feed on which no specific decontaminating step was applied) to 0% (for feed in which organic acids were added and a heat treatment was implemented). If contamination occurred, the level of contamination ranged from a few Salmonella kg(-1) feed to a maximum of 8E+04 Salmonella kg(-1) feed. Probability and levels of contamination were highest when no production process able to reduce or eliminate the pathogen was implemented. However, most of the Swiss production was shown to undergo some kind of decontaminating step. A heat treatment, in combination with the use of organic acids, was found as a solution of choice for the control of Salmonella in feed.

Animal Feed↗

PubMed2004

Development of predictive modelling approaches for surface temperature and associated microbiological inactivation during hot dry air decontamination.

This research deals with the development of predictive modelling approaches in the field of heat transfer and microbial inactivation. Upon making some backstage microbiological considerations, surface temperature predictions during hot dry air decontaminations are incorporated in a microbial inactivation model, in order to describe inactivation kinetics under realistic (time-varying) temperature conditions. In the present study, the following parts are presented. (i) First, a one-dimensional heat transfer model is developed taking into account exchanges by convection, radiation and evaporation. The model is subsequently validated on a laboratory setup and on a test rig, assuming no water activity changes. This test rig is developed for studying-at a later stage-surface pasteurisation treatment on food products with the use of hot dry air. (ii) Isothermal inactivation data of Escherichia coli K12 MG1655 have been collected and inactivation parameters are accurately estimated by using a primary and a secondary model in a global modelling approach. (iii) Microbiological considerations such as microbial growth effects during come-up times, initial temperature of inactivation, and heat resistance effects, based on experimental observations and on literature studies, are formulated in order to evaluate possible microbial effects arising under the dynamic temperature conditions modelled in step (i). (iv) Microbial inactivation simulations with the incorporation of surface temperature predictions are presented. (v) Finally, the level of the microbial decontamination in an example based on the design of an industrial installation is presented, outlining the importance of the combination of surface temperature and microbial inactivation modelling approaches.

Escherichia coli↗

PubMed2004

Modelling the individual cell lag phase: effect of temperature and pH on the individual cell lag distribution of Listeria monocytogenes.

The individual-based approach of the lag phase is gaining interest, especially for pathogens that initially contaminate food products in low amounts. In this paper, the effect of temperature (30, 10, 7, 4 and 2 degrees C) and pH (7.4, 6.1, 5.5, 5.0, 4.7 and 4.4) on the individual cell lag phase of Listeria monocytogenes was examined in a factorial design, using OD measurements. Individual lag phases of about 100 individual cells per condition were examined and calculated using a linear extrapolation method. Generation times were calculated out of the slope. The obtained data were analyzed at three different levels: in a first approach, the mean values were calculated for each set of environmental conditions and compared to predictions made by the USDA's Pathogen Modeling Program (PMP) for analogous growth conditions. The PMP predictions of the generation times were in the same order of magnitude as the obtained data, although a persistent underestimation could be observed. The observed individual cell lag data differed from lag phase predictions by PMP. Possible reasons for this discrepancy are discussed. Secondly, histograms of individual lag phase measurements were constructed for the different temperature-pH combinations. In this way, the influence of both factors on the variability of individual lag phases could be estimated. At low stress levels, most individual cells showed a short lag phase resulting in a compression of the histograms at the zero-lag level, while, at high stress levels, the histograms shifted to longer lag phases with a significant increase in variability. Thirdly, 37 different distribution types were fitted to the datasets to reveal the distributions that fitted best the obtained data. The gamma distribution was preferred at moderate stress levels, while the Weibull distribution was chosen for harsher growth conditions.

Colony Count, Microbial↗

PubMed2004

Predictive modelling of the microbial lag phase: a review.

This paper summarises recent trends in predictive modelling of microbial lag phenomena. The lag phase is approached from both a qualitative and a quantitative point of view. First, a definition of lag and an analysis of the prevailing measuring techniques for the determination of lag time is presented. Furthermore, based on experimental results presented in literature, factors influencing the lag phase are discussed. Major modelling approaches concerning lag phase estimation are critically assessed. In predictive microbiology, a two-step modelling approach is used. Primary models describe the evolution of microbial numbers with time and can be subdivided into deterministic and stochastic models. Primary deterministic models, e.g., Baranyi and Roberts [Int. J. Food Microbiol. 23 (1994) 277], Hills and Wright [J. Theor. Biol. 168 (1994) 31] and McKellar [Int. J. Food Microbiol. 36 (1997) 179], describe the evolution of microorganisms, using one single (deterministic) set of model parameters. In stochastic models, e.g., Buchanan et al. [Food Microbiol. 14 (1997) 313], Baranyi [J. Theor. Biol. 192 (1998) 403] and McKellar [J. Appl. Microbiol. 90 (2001) 407], the model parameters are distributed or random variables. Secondary models describe the relation between primary model parameters and influencing factors (e.g., environmental conditions). This survey mainly focuses on the influence of temperature and culture history on the lag phase during growth of bacteria.

Bacteria↗