S. Jaloustre
1*, M. Cornu
1, E. Morelli
1, V. Noel
1, and M.L. Delignette-Muller
21 Laboratoire d'Etudes et de Recherches sur la Qualité des Aliments et sur les Procédés agro-alimentaires, Agence française de sécurité des aliments, 23 avenue du Général de Gaulle, 94706 Maisons-Alfort, France.
(s.sevrin@afssa.fr)
2 Université de Lyon, CNRS UMR 5558, Laboratoire de Biométrie et Biologie Evolutive, Ecole Nationale Vétérinaire de Lyon, 1 avenue Bourgelat, 69280 Marcy l‟Etoile, France
* corresponding author
Introduction
Clostridium perfringens is a pathogen commonly found in meat products and often responsible for foodborne diseases in institutions and restaurants (Crouch and Golden, 2005).
These outbreaks are often due to spores germination and vegetative cells growth during a too slow chilling. Many studies have been published about modelling C. perfringens vegetative cells growth during chilling (Juneja et al., 1999, Juneja et al., 2001, Huang, 2004, Juneja et al., 2008, Le Marc et al., 2008). The aim of this study is to estimate the parameters of a growth model using all the data published to date on beef, to be able to predict C. perfringens growth in beef for risk assessment, taking into account potential sources of variability and uncertainty.
Materials and methods Growth data
Twenty five published growth curves were used from 6 experimental works published from 1980 to 2004. Each growth curve was obtained under constant thermal conditions (from 15°C to 50°C) after a heat shock (most often 75°C for 10 or 20 minutes), from one strain or a cocktail of C. perfringens strains inoculated in beef without modification of its natural composition. We directly used raw data, reported as measurement times (units : hour) and log counts (units : log10 cfu.g-1) values.
Growth model
The dataset made up of all the growth curves was modelled simultaneously by a Bayesian approach such as the ones proposed by Pouillot et al. (2003) and Delignette-Muller et al.
(2006). Stochastic and logical links between nodes are displayed on the directed acyclic graph (figure 1) and defined in table 1.
Growth curves were described by the primary growth model of Baranyi and Roberts (1995) with immediate transition between exponential and stationary phase (Eq.1). The effect of temperature on maximum specific growth rate was described by the cardinal temperature model of Rosso et al (1995)(Eq.2). Only the effect of the temperature on the specific growth rate was explicitly modelled. The effects of the other factors were taken into account by the value of the optimal specific growth rate (
optbeef). The cardinal temperatures (Tmin,Topt,Tmax) and the optimal specific growth rate (
optbeef) were not supposed variables. Quantity
0ln
h (equal to the product of the lag time and the maximum specific growth rate in constant environmental conditions) was used to describe the “work to be done” during the germination-outgrowth-latence phase (Baranyi and Roberts, 1995). It was supposed variable between growth curves and described by a normal distribution N(lnh0
,lnh0
). Input parameters of the model (table 2, figure 1) were considered as random and uncertain variables. Their prior distributions were defined from published data, that were not used in the computations (table 2). The empirical posterior distribution of each parameter was computed from its prior one and from the whole dataset. Computations were performed with JAGS.Growth curve c denoted by rectangles; covariates are denoted by double-rectangles and parameters are denoted by ellipses. Arrows run between nodes from their direct influence („parents‟) to the „descendants‟. Solid arrows indicate stochastic dependences while dashed arrows indicate logical functions. Stochastic and logical links are presented in table 1.
Table 1 : links between nodes of the growth model
Node Type Definition
Ym stochastic N(My,m,y) data of final cell number increase corresponding to several temperature scenarios were collected. A first personal dataset was obtained under constant temperature (6 hours at 45°C) but after a non classical heat shock chosen to mimic the cooking of beef-in-sauce products (increase from 10°C to 100°C in 40 minutes). Other datasets were obtained from published studies where the heat shock was classical but followed by an exponential cooling from 54°C to 7°C in 12 to 21 hours, depending on the study. For each temperature profile, a set of 1500 values of the growth model parameters (table 2) was randomly selected from the joint posterior distribution resulting from the Bayesian modeling, so as to take into account uncertainty on each input of the model in the simulations.
Results and discussion Growth model parameters
Statistics of the empirical distribution of the uncertain parameters are presented in Table 2.
For most of these parameters, Bayesian inference succeeded in narrowing distributions, thus giving rather precise estimations, except for Tmax, due to the lack of growth kinetics obtained
above 50°C. Estimated parameters of the normal variability distribution of
ln
h0 (lnh0andlnh0
) are reflecting the great variability of germination-outgrowth-latence phase duration observed in the data.As shown on figure 2, the model seems to reasonably describe the observed values, without clear outlier.
Table 2: Prior distributions (defined from previous publications cited below the table) and empirical posterior distributions of the inputs of Clostridium perfringens growth model.
Parameter Prior distributions Posterior distributions
Defined distributions 2.5th percentile
97.5th percentile
Median 2.5th percentile
97.5th percentile
Tmin a N(10,2) 6 14 12.7 12.1 13.1
Topt b N(44,2) 40 48 45.1 44.2 46.2
Tmax a N(52,1.5) 49 55 53 51.5 55
optbeef
c N(4.5,1.8) 1.1 8 3.8 3.5 4.2
lnh0
d N(1.95,0.32) 1.3 2.6 1.8 1.4 2.1
lnh0
e 2 (0.001,0.001) 0
lnh Gamma
1.5 x 105 ∞ 0.9 0.6 1.2
ye y2Gamma(0.001,0.001) 1.8 x 104 ∞ 0.33 0.3 0.36
a Blankenship et al. (1988), Juneja et al. (1994), Juneja et al. (2002), de Jong et al. (2005), Le Marc et al. (2008) b de Jong et al. (2005)
c Schroder et al. (1971), Willardsen et al. (1978), Willardsen et al. (1979), Blankenship et al. (1988), Labbe et al. (1995), Juneja et al. (2002), Jong et al. (2005), Le Marc et al. (2008), Juneja et al. (2008)
d Le Marc et al. (2008)
e classical non informative priors on precision parameters
Fig. 2 : Comparison of observed log counts with the averaged predicted ones.
Validation results
Figure 3 represents simulated 95% credibility intervals and observed values of final cell number increase under different temperature scenarios. The credibility intervals reflect the overall uncertainty combining variability and uncertainty. This uncertainty seems great but is for most part due to variability on
ln
h0 and is consistent with the variability observed on the validation data, especially when data come from various authors, which is the case for scenarios d and e.Discussion
Cardinal temperatures and optimal growth rate were not considered as variable in the proposed model due to the lack of available data enabling the characterization of this
variability. It would be of interest to explore in particular the inter strain variability by individually culturing various strains.
In our modeling approach, different alternatives were compared for parameter
ln
h0 , first assumed not variable, then assumed variable between studies and at last variable between curves. The last alternative was chosen for its far better description of data. The source of this observed variability seems hard to biologically explain without any further investigation, as data collected for Bayesian modeling were all obtained on uncured beef, mostly with the same classical heat shock and with only one strain or cocktail for each study. Nevertheless, the impact of this variability on the overall uncertainty of growth model predictions should be taken into account in risk assessment.Fig. 3 : Simulated intervals and observed values of final cell number increase under different temperature scenarios (a: cooking-like heat shock followed by 6 hours at 45°C, b to e : classical heat shock followed by exponential cooling from 54°C to 7 °C within 12 hours (b), 15 hours (c), 18 hours(d), 21 hours (e)). Dark lines represent the 95% credibility interval obtained by simulation, while crosses represent observed values. Data corresponding to scenarios b to e were collected from Thippareddi et al. (2003), Smith et al. (2004), Sanchez Plata et al. (2005), Juneja et al. (2006), Juneja et al. (2007), Juneja et al (2008)
Acknowledgements
We would like to thank the ANR for providing financial support, and the COMBASE and the SYM‟PREVIUS projects for their contribution by the availability of the data.
References
Baranyi, J., Roberts, T.A. (1995). Mathematics of predictive microbiology. International Journal of Food Microbiology 26 : 199-218.
Crouch, E. and Golden, N. (2005) A Risk Assessment for Clostridium perfringens in Ready-To-Eat and Partially Cooked Meat and Poultry Products, USDA, Food Safety Inspection Service, September 2005.
Delignette-Muller, M.L., Cornu, M., Pouillot, R. and Denis, J.B. (2006) Use of Bayesian modelling in risk assessment: Application to growth of Listeria monocytogenes and food flora in cold-smoked salmon.
International Journal of Food Microbiology 106, 195–208
Juneja, V. K., Whiting, R. C., Marks, H. M. and Snyder, O. P. (1999) Predictive model for growth of Clostridium perfringens at temperatures applicable to cooling of cooked meat. Food Microbiology 16, 335-349
Juneja, V.K., Novaka, J.S., Marksb, H.M. and Gombas, D.E. (2001) Growth of Clostridium perfringens from spore inocula in cooked cured beef: development of a predictive model. Innovative Food Science & Emerging Technologies 2, 289-301
Juneja, V.K., Marks, H. and Thippareddi, H. (2008) Predictive model for growth of Clostridium perfringens during cooling of cooked uncured beef. Food Microbiology 25, 42–55
Huang, L. (2004) Numerical analysis of the growth of Clostridium Perfringens in cooked beef under isothermal and dynamic conditions. Journal of Food Safety 24, 53-70.
Le Marc, Y., Plowman, J., Aldus, C.F., Munoz-Cuevas, M., Baranyi, J. and Peck, M.W. (2008). Modelling the growth of Clostridium perfringens during the cooling of bulk meat. International Journal of Food Microbiology 128, (1) 41-50.
Pouillot, R., Albert, I., Cornu, M. and Denis, J.-B. (2003). Estimation of uncertainty and variability in bacterial growth using Bayesian inference. Application to Listeria monocytogenes. International Journal of Food Microbiology 81(2): 87-104.
Rosso, L., Lobry, J.R., Bajard, S., Flandrois, J.P. (1995) Convenient Model To Describe the Combined Effects of Temperature and pH on Microbial Growth. Applied and Environmental Microbiology 61 (2), 610–616