This model was first proposed by Gibson et al. (1988) to describe microbial growth curves and reparameterized by Zwietering et al. (1990) in order to define it with classical growth parameters N0, Nmax, mmax, and 1. But the biological interpretability of its parameters is not so obvious since Eq. 21.10 does not describe a value of N equal to N0 at time t = 0 and since the model is a logistic one with mmax representing the slope of the tangent of the curve at the inflection point and not the constant slope of the curve during the exponential phase as expected.
When these primary models are used to estimate growth parameter from kinetics data, they are generally fitted by nonlinear regression to data points transformed in logarithm (on log10(N) or ln(N)), using an additive Gaussian error model. This can be done using any statistical software providing a nonlinear regression function or using specific software. For example, DMFit is an Excel add-in freely provided by the Institute of Food research on its web site (http://www.ifr.ac.uk/safety/DMFit/default.html) that enables the fit of growth kinetics using the Baranyi model or the three-phase linear model or simplified versions of both the models, assuming, for example, that there is no lag phase or that stationary phase was not reached. In the R package for nonlinear regression diagnostics (nlstools, 2007), the same models are proposed with the modified Gompertz one in addition. This package may be directly installed from the free R language (R Development Core Team, 2007) or downloaded from the web site of R foundation for Statistical Computing (http://lib.stat.cmu.edu/R/CRAN/) and used with the R language. Figure 21.2
Fig. 21.2 Fitting of three classical primary growth models on the same growth data set t
Fig. 21.2 Fitting of three classical primary growth models on the same growth data set shows the three models fitted on the same growth data set using functions provided by nlstools.
When these models are compared on different data sets, none of them systematically gives a better fitting for all the data sets. They all seem to fit data sets quite well even if they do not give exactly the same estimations of growth parameters (McKellar & Lu, 2004). The modified Gompertz model, due to its logistic form, is known to overestimate the maximum specific growth rate in comparison to other models and tends to give more imprecise estimations of growth parameters for kinetics with few data points (Baty & Delignette-Muller, 2004). Moreover, the Baranyi model and the three-phase linear model are more flexible, with simplified versions for the cases where there is no lag phase or where the stationary phase is not reached at the end of the kinetics, and they both have a differential formulation which permits their use in changing environmental conditions.
The classical model still often used to describe survival kinetics is the log-linear model:
where N is the cell concentration and kmax the maximum specific inactivation rate. The second parameter kmax is directly linked to the classical decimal reduction time D by D = ln( 10)/kmax, often named the D-value. But microbial survival kinetics, whether due to a thermal or a nonthermal food processing, may present various forms differing from the linear one. Geeraerd, Valdramidis, and Van Impe (2005) described nine typical forms characterized by a concavity or convexity and the possible presence of a tail and/or a shoulder and reviewed the various primary models proposed by different authors to describe all the possible forms (Geeraerd, Herremans, & Van Impe, 2000). Some of these models are adaptations of the Baranyi growth model, considering a survival curve as a mirror image of a growth curve. The model proposed by Geeraerd et al. (2000) is of this type and is classically parameterized by N0 the initial cell concentration, Nres the residual cell concentration at the end of the kinetics, kmax the maximum specific inactivation rate, and Si the shoulder (in time):
a(t) = 1 + C (t); with Cd- = -kmax, and Cc(0) = exp(kmaxSi) - 1 (21.13) and
Using the same analogy between a survival curve and a growth curve, it is also easy to define a three-phase linear model characterized by the same four parameters by Eq. 21.12 with Eqs. 21.15 and 21.16:
Some authors also use models derived from the Weibull model first proposed by Peleg and Cole (1998) and adapted by other authors (Mafart, Couvert, Gaillard, & Legu^rinel, 2002; Albert & Mafart, 2005). The initial Weibull model is based on the assumption that the time required to cause the death of one cell is variable in the microbial population and follows a Weibull distribution. The last model proposed by Albert et al. is parameterized by four parameters: N0 the initial cell concentration, Nres the residual cell concentration at the end of the kinetics, d the time of the first decimal reduction, a parameter close to the classical D-value, and p a shape parameter, characterizing the curve convexity or concavity. This model can be written in a similar form as the two others, by Eqs. 21.12, 21.14 and 21.17:
As growth models, survival models are generally fitted by nonlinear regression to data points transformed in logarithm (on log10(N) or ln(N)), using an additive Gaussian error model. Geeraerd et al. (2005) developed an Excel add-in named Ginafit, which is freely provided by the University of Leuven (http://cit.kuleuven.be/biotec/). It enables the fit of survival kinetics using nine different models, among which the Geeraerd model and Albert model previously presented their simplified versions assuming for example that there is no shoulder or no tail. In the R package nlstools, the fit of eight models, the three models previously described, and their simplified versions are proposed. Figure 21.3 shows these three models fitted on the same survival data set using functions provided by nlstools.
The Geeraerd model and the three-phase linear model give generally close estimations of the four classical parameters N0, Nres, kmax and Si. The simple interpretability of their parameters is an obvious quality of these models. Nevertheless, they do not enable the fit of all the survival curves. In fact, the form of survival curves is very variable and dependent on the type of food processing. In this context, the Weibull-type models (Mafart et al., 2002; Albert & Mafart, 2005) are sometimes preferred in predictive modeling works for their better fit to the data (Fernandez, Lopez, Bernardo, Condon, & Raso, 2007; Janssen et al., 2007; Leguerinel et al., 2007).
Secondary models are developed to describe the effect of environmental factors on behavior of microorganisms. They are generally developed for one microbial species, from experiments made on one or various strains of the species. Most of them describe the effect of environmental parameters directly on the primary growth or survival parameters. In order to develop such models, it is thus necessary to measure many growth or survival kinetics in different
Fig. 21.3 Fitting of three classical primary survival models on the same survival data set environmental conditions characterized by one or more factors. The experimental design is of great importance at this stage (van Boekel & Zwietering, 2007). After data collection, it is necessary to fit a primary model on each of those kinetics, then to fit a secondary model to the estimated values of these parameters as a function of the environmental factors. A global fit of the whole data set is also possible and even preferable from a statistical point of view, but needs a greater statistical skill. As primary models, secondary models should be parsimonious, with no more parameters than are required to describe the data (Baranyi, Ross, McMeekin, & Roberts, 1996) and if possible easily interpretable parameters (Ross & Dalgaard, 2004).
In this chapter, we will first focus on the secondary modeling of the specific growth rate, developing some of the various approaches proposed. Then we will introduce the modeling of the lag time, the modeling of inactivation parameters, and the modeling of the probability of growth.
Secondary Models for Growth Rate Models Based on the Gamma Concept
Many models are based on the square root model proposed by Ratkowsky et al. (1982) to describe the effect of the temperature T on the maximum specific growth rate ^max:
where b is a constant and Tmin the theoretical minimum growth temperature. Different models were proposed to extend the applicability of this model to temperatures near and above the optimal growth temperature (Ratkowsky, Lowry, McMeekin, Stokes, & Chandler, 1983; Zwietering, de Koos, Hasenack, de Wit, & van't Riet, 1991; Rosso, Lobry, & Flandrois, 1993). Among them the model developed by Rosso et al., named the cardinal temperature model, is one of the more often used models. It offers a great advantage to be characterized by parameters which have an obvious biological or graphical interpretability. These parameters are the minimum growth temperature Tmin, the maximum growth temperature Tmax, the optimal growth temperature Topt, and the optimal specific growth rate mopt reached at this temperature. This commonly used model belongs to the family of the cardinal parameter models (Rosso and Robinson, 2001) and can be defined by Eqs. 21.19 and 21.20 as follows:
(Xopt-Xmin) [(Xopt—Xmin ) (X—Xopt ) — (Xopt-Xmax ) ((n—1)Xopt+Xmin-nX)] '
Xmin 5X5 Xmax (21.20
In this family the cardinal pH model (Rosso, Lobry, Bajard, & Flandrois, 1995) defined by Eq. 21.21 and the cardinal aw model defined by Eq. 21.22 were also developed:
Each of these models is often simplified as data points near the maximum parameter value are rare; in the cardinal pH model, the maximum growth pH (pHmax) is often fixed to 2pHopt — pHmin and in the cardinal model the parameter awopt is often fixed to 1.
For the simultaneous modeling of the maximum specific growth rate mmax as a function of more than one environmental factor, the gamma concept first introduced by McMeekin et al. (1987) and named by Zwietering, Wijtzes, de WIT, and van't Riet (1992) is often used. The gamma concept relies on the observation that in many conditions, environmental factors act independently on mmax and that the global effect may be described by a multiplicative model. For example, a model describing the simultaneous effect of T, pH, and may be simply written from previous equations as mmax(T) = mopt X CM2(T) X CM^pH) X CM2K) with max = 1 (21 .23)
The gamma concept has been extended to take into account interactive effects between factors in conditions where such interactive effects are observed (Augustin & Carlier, 2000; Le Marc et al., 2002), but simple models based on the gamma concept are often sufficient to correctly describe the simultaneous effect of environmental factors (Lambert & Bidlas, 2007).
Models for specific growth rates are generally fitted by nonlinear regression to data points transformed in square root (on pmmax), using an additive Gaussian error model (Ratkowsky, 2004). In the R package nlstools, six models are proposed: the cardinal temperature model, the cardinal pH model in its two forms (three or four parameters), the cardinal model in its two forms, and the complete model proposed by Pinon et al. (2004) with nine parameters (mopt, Tmin, Topt, Tmax, pHmin, pHopt, pHmax, aw min and awpt). As an example of fitting of such a model, Fig. 21.4 represents the fitting of the cardinal temperature model to a data set published by Tamplin, Paoli, Marmer, and Phillips (2005) concerning the effect of temperature on the growth of E. co/z O157:H7 in raw sterile ground beef, using nlstools functions.
In the past, polynomial models were very often used to describe the simultaneous effect of various environmental factors. They were extensively used during the 1990 s and remain widely applied although models based on the gamma concept are now becoming popular. This extensive use is certainly due to the fact that polynomial models are very easy to fit by multiple linear regression, available in most statistical packages. The mathematical form of polynomial models is always the same and theoretically enables the fit of any data sets corresponding to various values of growth or inhibition parameters observed for different values of environmental factors. For example, the natural logarithm of the generation time, ln(tg), was often modeled as a quadratic
Fig. 21.4 Fitting of the cardinal temperature model to a data set published by Tamplin et al. (2005) concerning the effect of temperature on the growth of E. coli O157:H7 in raw sterile ground beef
Fig. 21.4 Fitting of the cardinal temperature model to a data set published by Tamplin et al. (2005) concerning the effect of temperature on the growth of E. coli O157:H7 in raw sterile ground beef polynomial function of environmental factors (for example, T the temperature, P the pH and S the NaCl percentage) by the following equation:
ln(ig) = b0 + b1T + b2 P + b3S + b4T2 + b5P2 + b6S2 + b7TP + b8TS + b9PS
Such a model, also called a surface response model, is easy to fit to experimental data, whatever the number of environmental factors, and the fit can be easily improved when necessary by increasing the order of the polynomial equation. But the number of parameters to estimate from the data (b0,b1;...,) rapidly increases with the number of environmental factors and with the order of the polynomial equation. In the past, cubic models with 35 parameters were sometimes proposed to predict the generation time as a function of 4 environmental factors (Buchanan & Bagi, 1994). Such flexible models have been criticized, as they attempt to model the experimental error rather than eliminate it, and thus lack robustness (Baranyi et al., 1996). Their use should be restricted to a fit on huge data sets of very high quality and a predictive use only on the interpolation region of the experimental plan, which is smaller than the intuitive rectangular parallelepiped defined with the ranges of environmental factors, and not so easy to characterize (Baranyi et al., 1996). An approach was proposed by Geeraerd et al. (2004) to avoid drawbacks due to a too great flexibility, by developing polynomial models respecting biologically predefined constraints. But the fit of models with such an approach is no more easy, and thus polynomial models seem to lose their main quality.
Another drawback of polynomial models is that their parameters are coefficients (b0, b1,..., bk) which are not easily interpretable. They have no direct biological meaning in contrary to parameters such as the minimum growth temperature Tmin. The result of the fit of a polynomial model on a data set may thus be really difficult to interpret. For example, the statistical significance of interaction coefficients (b7, b8, or b9 in Eq. 21.24) is often interpreted as the proof of a biological interaction between corresponding factors, as it may be caused only by a bad global fit of the model to the data set (Lambert & Bidlas, 2007). As another consequence of the lack of interpretability of model parameters, different polynomial models are very difficult to compare.
In conclusion, due to the drawbacks previously described, the use of polynomial models should be restricted to cases where no other approach is possible, and the development and use of such models should be very cautious (Ratkowsky, 2004; Ross & Dalgaard, 2004).
The previous presentation of existing models is not exhaustive. Some of the more commonly used models were presented, but many other models were proposed. Other models derived from the square root models were developed based on the gamma concept as the cardinal parameter models. Models derived from the Arrhenius equation were also proposed, but far less used than other models. The use of neural networks was also proposed to develop secondary models, but in this "black box'' approach, the same drawbacks than those of polynomial models are found again. A complete review of secondary models may be found in the book ''Modelling Microbial Responses in Food" edited by McKellar and Lu, in the corresponding chapter (Ross & Dalgaard, 2004).
Other Secondary Models Secondary Models for Lag Time
The lag time l corresponds to the time needed by bacterial cells to adapt to a new environment, for example after food contamination, before starting an exponential growth. This parameter is much more difficult to predict than the maximum specific growth rate mmax, as it does depend not only on current conditions but also on previous environmental conditions of the bacterial cells and on their physiological state. Many authors have reported a strong influence of the pre-incubation temperature on the lag phase duration (Swin-nen, Bernaerts, Dens, Geeraerd, & Van Impe, 2004). As an example, bacterial cells previously cultured at low temperatures have a reduced lag at low temperatures compared with cells previously cultured at high temperatures (Membre, Ross, & McMeekin, 1999; Whiting & Bagi, 2002).
During the 1990s, many authors modeled the lag time independently of the maximum specific growth rate (or generation time) (for a review, see Delignette-Muller, 1998). They generally proposed polynomial l models developed from growth kinetics of cells previously cultured at a favorable high temperature. Consequently, when these models are used to predict the growth of an environment contaminant in a refrigerated food product, l is overestimated. Membre et al. (1999) suggested that in such studies, microorganisms should be previously cultured at low temperatures, in order to mimic the processes of contamination in industry.
Other authors assumed that the product h0 = mmax1 does not depend on the growth conditions, but only on the pre-incubation conditions. Under this assumption, l may be simply predicted from the predicted value of mmax and from the constant h0 for given pre-incubation conditions (Augustin & Carlier, 2000; Pinon et al., 2004). This product h0 was described as the "work to be done'' by the cells during the lag phase to prepare for the exponential growth (Robinson, Ocio, Kaloti, & Mackey, 1998; Pin, Garcia de Fernando, Ordonez, & Baranyi, 2002). But this product may be considered constant only in a first approximation (Delignette-Muller, 1998; Pin et al., 2002), and authors showed an increase of its value with the magnitude of the shift between two environmental conditions (Mellefont & Ross, 2003; Delignette-Muller et al., 2005; Mellefont, McMeekin, & Ross, 2005).
Another difficulty encountered while trying to develop predictive models for the lag time is that the observed population lag time depends on the inoculum level. The population lag time decreases with the initial number of cells for law inocula, such as those that may be encountered in realistic food contaminations. As this observed effect is due to the variability among individual cell lag times (Baranyi, 1998; Augustin, Brouillaud-delattre, Rosso, & Carlier, 2000), a predictive model should take account of this variability by describing the distribution of individual lag times. Models were recently proposed to describe the distribution of individual lag times and the effect of environmental factors on this distribution (Francois et al., 2005; Guillier & Augustin, 2006; Standaert et al., 2007), but these models still need to be compared and validated before an extensive use.
In conclusion, considering the modeling of environmental factors on the lag time, more research is still required before secondary models as predictive as the growth rate models can be proposed for lag time.
The Bigelow model has been a standard for decades to predict the effect of temperature on the classical decimal reduction time D (D-value), and this model was also more recently used to model in the same way the time of the first decimal reduction, d, estimated from the fit of the Weibull inactivation model (Mafart et al., 2002). The Bigelow model simply describes the logarithm of the ratio between the D-value at the temperature T and the D-value at a reference temperature Tref as a linear function of the difference T — Tref by the equation
The parameter z corresponds to the reciprocal of the slope and is commonly named the z-value. Such a model may be very easily fitted by simple linear regression on the logarithm of the D-values. This model has been used for decades to calculate processing times for thermal processes such as pasteurization and sterilization.
Only few models were developed to predict the effect of multiple environmental factors on inactivation parameters, and most of them are polynomial models (Ross & Dalgaard, 2004). Some attempts were made to extend the Bigelow model by describing the effect of other environmental factors on Weibull inactivation parameters using a progressive approach such as the one proposed in the gamma concept (Couvert, Gaillard, Savy, Mafart, & Leguer-inel, 2005; Leguerinel et al., 2007). More research is needed in its field in order to propose predictive models taking into account factors other than temperature that may contribute to inactivation.
Several types of models were proposed to describe the growth/no growth limit as a function of various environmental factors. A deterministic modeling of this limit may be deduced from secondary growth models based on the gamma concept (Lambert & Bidlas, 2007) from the characterization of parameters such as the minimum growth temperature (Tmin), taking into account if necessary interaction between environmental factors (Augustin & Carlier, 2000; Le Marc et al., 2002). A stochastic modeling of this limit may also be performed using logistic regression to model categorical data (growth or no growth) as a function of various environmental factors. This use of logistic regression in predictive microbiology was introduced by Ratkowsky and Ross (1995) and used by many authors.
In logistic regression, the logit function of the probability of growth (logit(p) = ln(p/(1 — p))) is described as a function of the environmental factors. Two types of models are commonly used: the first describing logit(p) as a polynomial function such as the one of Eq. 21.24 and the second describing
logit(p) as a nonlinear function based on models developed using the gamma concept (Ross & Dalgaard, 2004; Gysemans et al., 2007). The second approach should be preferred when such models give reasonable fits to data, for the reasons already developed when dealing with mmax models. It gives more robust models, which are parameterized with interpretable parameters (Gysemans et al., 2007). Nevertheless, the first approach with polynomial functions might be used for data sets showing nontypical growth limits, such as those observed by Gysemans et al. (2007) on mixed strain data.
As for modeling the lag time, the potential effect of the past of the microbial cells and of the inoculum should be considered in experimental works aimed at characterizing the growth/no growth limit. As for the lag time, the variability among individual cells may have a greater impact on this limit for low inocula (Skandamis et al., 2007). Indeed, at a given environmental condition, the larger the cell population, the more likely it is to contain at least one cell that is capable of initiating growth. Concerning the potential impact of the past of the cells, Skandamis et al. (2007) reported an impact of a preliminary acid adaptation of E. co/z O157:H7 cells on their growth/no growth limit, but further researches are needed before models could take such impacts into account.
Applications of Predictive Modeling
The Challenge of Modeling Microbial Dynamics in Food Under Realistic Conditions
Models developed from microbial experiments in laboratory conditions must be validated in realistic conditions before being used as predictive models. Validation studies must demonstrate that microorganisms in food products behave in a similar way as in laboratory conditions. Users of such models must be aware of the limitations of their applicability.
The models previously described were often developed from data obtained from a culture of one or more strains of one species, in liquid media, and in constant environmental conditions. To be able to predict survival and growth of microorganisms in realistic conditions, these models must be extended to take into account the effect of changing environmental conditions, of the potential interaction between various strains from different species developing in a same food product and of the effect of the structure of nonliquid food products.
From a mathematical point of view, it is not difficult to predict microbial growth with changing environmental conditions, by numerically integrating the differential equations defining the growth model, as first proposed by Baranyi, Robinson, Kaloti, and Mackey (1995). This is generally done assuming that the primary model parameters (mmax, for example) immediately change according to the changing environmental factors and the secondary model. This hypothesis seems to be reasonable to predict microbial growth under nonisothermal conditions, as far as the thermal scenario is sufficiently smooth, but delayed responses may be observed for sudden strong fluctuations of the temperature (Baranyi et al., 1995; Swinnen, Bernaerts, Gysemans, & Van Impe, 2005). The use of this procedure has been validated several times for food products under realistic nonisothermal conditions, especially in meat products (Mataragas, Drosinos, Siana, Skandamis, & Metaxopoulos, 2006; Koutsoumanis, Stama-tiou, Skandamis, & Nychas, 2006). The same type of procedure may be used to predict microbial inactivation, using the same biological hypothesis. Its use was not so extensively validated than for growth prediction. It was recently successfully validated for nonisothermal realistic conditions (Aragao, Corradini, Normand, & Peleg, 2007) and high-pressure-changing conditions (Koseki & Yama-moto, 2007), but it was shown by other authors that this procedure may overestimate the effect of an applied heat treatment as not taking into account the potential physiological adaptation of the microorganisms (Bernaerts et al., 2004; Valdramidis, Geeraerd, & Van Impe, 2007).
Concerning the potential effect of the natural flora of the food product and of its structure, some attempts were made to model them in some specific cases. The interaction between food flora and a pathogen microorganism may, in some cases, be modeled by a simple competition between both populations (Vimont et al., 2006; Mellefont, McMeekin, & Ross, 2008), but other more complex interactions may also be observed in food. A few models were proposed to describe growth interactions involving the diminution of the pH or the production of bacteriocins due to microbial growth (Janssen et al., 2006; Leroy & De Vuyst, 2007), but much work still needs to be done before proposing a global modeling approach of microbial interactions. Concerning the effect of food structure, a first model was recently proposed, but much work still needs to be done on this subject too (Antwi, Bernaerts, Van Impe, & Geeraerd, 2007). As very complex phenomena might occur during microbial growth in food products, one should keep in mind that a preliminary validation of the use of models in the food product of interest and in realistic environmental conditions is required to ensure reasonable predictions.
The use of predictive models is of great interest to improve the microbiological safety of food products, but among the large number of published models, only a minority of them is electronically accessible from simulation tools. For the other models, it is always possible to develop its own simulation tool from the publications, but this requires some more time and expertise from the user. Some simulation tools in predictive microbiology are freely provided. The
Pathogen Modeling Program (PMP) has been developed by the US Department of Agriculture-Agricultural Research Service (USDA-ARS) since the 1990 s. It includes models for growth and inactivation and is updated from time to time to include newly developed models. It may be freely downloaded from the USDA-ARS web site (http://ars.usda.gov/services/software/software.htm). The Growth Predictor is the successor to the Food MicroModel program and has been developed by the UK Institute for Food Research. It provides only growth models as indicated by its name. It may be freely downloaded from the UK Institute of Food Research web site (http://www.ifr.ac.uk/Safety/Growth Predictor). The Pathogen Modeling Program and the Growth Predictor include models that were developed in broth media and not necessarily validated in food products. A preliminary validation of simulations in the food product of interest and in realistic environmental conditions is thus essential for a safe use. For this validation, the user may use its own experimental data and/or published data. The search for published data corresponding to a specific microbial species, a specific food product, and a set of environmental factors is facilitated by the use of the ComBase database. ComBase is a very large, freely available repository of microbiological data for predictive microbiology, accessible from the web (http://wyndmoor.arserrc.gov/combase/). Some few programs providing simulations validated on food products exist. The SeafoodSpoilage and Safety Predictor (SSSP) has been developed by the Danish Institute for Fisheries Research and is freely accessible to simulate the spoilage of seafood products (http://www.difres.dk/micro/sssp/). The Sym'Previus program (http://www.symprevius.org/) is a simulation tool that includes microbiological data from literature, published models, data from challenge tests in food products, but its use is not free and is restricted to registered users.
Time-temperature integrators or indicators (TTI) may be seen as another type of tool from quantitative microbiology. Such tools have been developed from the 1990 s in order to record the temperature with time and translate its effect on the microbial growth in the product of interest in easily readable information such as a color indicator. A TTI is a simple inexpensive device that indicates the temperature history in terms of microbiological status of the food product. Such devices may be based on mechanical, chemical, enzymatic, or microbiological systems that produce a color change as much rapidly as the temperature is high (Taoukis, Koutsoumanis, & Nychas, 1999; Vaikousi, Biliaderis, & Koutsoumanis, 2008). Applications of TTI to optimize the microbiological control of fish or meat products were reported (Moore & Sheldon, 2003; Koutsoumanis et al., 2006).
Quantitative microbiological risk assessment (QMRA) has become a classic approach in food microbiology, generally divided into four stages: hazard identification, hazard characterization, exposure assessment, and risk characterization. The use of models from predictive microbiology is often essential in the exposure assessment. Predictive models used in an exposure assessment should have been previously validated and the different sources of variability and uncertainty in these models should be clearly stated (Nauta, 2007). Indeed both uncertainty and variability may lead to imprecise model predictions, and each one should be characterized separately. Variability refers to true heterogeneity of the population considered and cannot be reduced by additional data, while uncertainty refers to lack of knowledge and may be reduced by the acquisition of additional data. Variability may correspond to variability between strains of the same species, that may have a great impact on the results of a QMRA (Delignette-Muller & Rosso, 2000), or variability in the composition of the food and in the growth conditions not included in the model as environmental factors (Delignette-Muller, Cornu, Pouillot, & Denis, 2006). Uncertainty may rely on the choice of the good model and on the estimation of the parameters of the chosen model, especially due to a lack of data. One should be aware that variability on predictive models' parameters is often underestimated, as microbiological practice tends to reduce sources of variability by standardizing experiments in order to reach reproducible results. New approaches, such as the use of Bayesian inference, may help to quantify the variability and uncertainty on parameters of predictive models from data from disparate sources (Pouillot, Albert, Cornu, & Denis, 2003; Delignette-Muller et al., 2006).
Much work has been done in predictive microbiology during the last 30 years but much work still needs to be done. Even if a great number of models have been published, a minority of them has been included in freely provided simulation tools, and only a small number of them have been clearly validated for predictions in food products under realistic conditions. Nowadays, a safe use of these models thus often requires a good background in modeling and the acquisition of some data on the food products of interest. An effort has been done by the researchers in predictive microbiology to facilitate the use of models and the access to microbiological data, but this effort should be maintained in order to make the use of models even more easy and safe. Moreover, some research fields in predictive microbiology, such as modeling of microbial interactions or modeling of the effects of food structure, have just emerged and should give interesting results in the next years. New trends may also emerge in predictive microbiology from links with other research fields such as genomics, bioinfor-matics, or systems biology.
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