A brief overview of the models for C. botulinum growth was discussed by McClure, Cole, and Smelt (1994). The earliest models developed for C. botuli-num were for predicting the survival of the spores during thermal processing (canning). Subsequent models for probability of growth or time to toxin production have been reported in the literature. The factors that have been used for preserving foods at risk for C. botulinum growth and toxin production include pH, NaCl, sodium nitrite, and sorbate among others. Predictive models describe the effects of these factors in a variety of food matrices such as cheese spreads, pork slurry (Lindstrom et al., 2001), and sous vide beef and sous vide pork (Hyytia-Trees et al., 2000).
Genigeorgis, Meng, and Baker (1991) used a two-step approach, using regression analysis to model lag phase, and subsequently incorporated these into secondary equations to express or predict the probability of toxin production, resulting from C. botulinum growth in turkey homogenate. A more user friendly approach to modeling, using kinetic models (primary and secondary models) with a variety of parameters aw, pH, NaCl, and nitrite utilizing meat products, would be more appropriate as models generated using microbiological media may not provide "realistic" estimates of C. botulinum growth.
Polynomial expressions that incorporated the environmental variables (pH, aw, NaCl, etc.) were used to develop probability models (logistic) to estimate the probability of toxin formation by C. botulinum (Roberts, Gibson & Robinson, 1981). Subsequently, Lindroth and Genigeorgis (1986) developed a probability model assuming germination, growth initiation, and toxin production from a single spore, using a similar expression.
Primary-secondary models use (i) either a kinetic model to describe lag time and growth of the organism or a probability model to predict the chance of toxin formation over time and (ii) another model to predict the effect of environmental factors on the parameters of the first model (Schaffner, Ross, & Montville, 1998).
Gibson et al. (1987) developed a kinetic model for the growth of C. botulinum type A in pasteurized pork slurry by using logistic and Gompertz functions. The relationship between the time to reach the maximum rate of growth and incubation temperature and sodium chloride concentration was described graphically.
Whiting and Call (1993) used nonlinear regression to estimate the parameters of a primary model for probability of growth at a given time and then used polynomial expressions containing experimental variables to predict the parameters of the primary model. This approach was expanded to develop a model for non-proteolytic type B C. botulinum, where inoculum size and time-to-toxicity confidence intervals were also included in the model (Whiting & Oriente, 1997).
While the modeling approaches for probability models and kinetic models are different, both the approaches basically predict the ability of the organism to grow and, in case of growth, subsequent production of toxin (Schaffner et al., 1998). Other approaches used to model C. botulinum behavior include waiting time modeling (Ter Steeg & Cuppers, 1995) to develop expressions for the effect of environmental parameters on time for a specific event (toxin production, turbidity development, or growth of the organism). Waiting time models can be used whenever the time to the occurrence of some event is the variable of interest. In the case of the time-to-toxicity data, this is the time from the beginning of an experiment until a tube is identified as positive. Schaffner et al. (1998) stated that waiting time models can be easily developed using currently available statistical analysis software; the models are flexible and are simple to interpret (Fig. 22.2).
Rogers and Montville (1994) used linear regression to model the factors that influence the ability of nisin to inhibit C. botulinum in a model food system. Subsequently, Schaffner et al. (1998) used the waiting time modeling approach to analyze the combined effects of temperature, pH, carbohydrate, protein, and lipid on the time-to-toxicity of C. botulinum using data from Rogers and Montville (1994). Fernandez, Baranyi, and Peck (2001) developed a model (quadratic, multivariate response surface) to predict the growth ofnon-proteolytic C. botulinum in a model system (PYGS broth) at various pH, NaCl, temperature, and CO2 concentrations (modified atmospheres).
While there were several studies indicating the time to toxin production or turbidity for C. botulinum, the minimum populations required to produce toxin have not been reported. Review of Elliott and Schaffner (2001) indicates that the C. botulinum populations were >5.0 log CFU/ml before toxin was detected
in TPGY broth containing NaCl (0.25 or 1.75%), at pH 5.75 or 6.5 and temperature of either 7 or 13°C. It would not be prudent to allow the growth of C. botulinum regardless of the ability of the organism to produce toxin in the product in question. The USDA-FSIS stabilization requirements specify no growth of C. botulinum during cooling of the meat and poultry products (USDA-FSIS, 1999). During cooling of meat and poultry products, process deviations can occur, resulting in abusive cooling rates (beyond the safe harbors), either at the higher temperature range (54.4-26.7°C) or the lower temperature range (26.7-4.4°C). In terms of C. botulinum spore germination and outgrowth, the potential for either proteolytic and non-proteolytic types or both should be evaluated based on the product temperature (range) where the deviation occurred and the cooling profile of the product.
Hyytia-Trees et al. (2000) evaluated the performance of the ''tertiary'' models for non-proteolytic C. botulinum (UK Food MicroModel; FMM and USDA-ARS Pathogen Modeling Program) growth in sous vide-processed products and reported significant variation between the safe storage time predictions from the software and the challenge study. The authors ascribed the poor agreement between the predictions from the software and the challenge study to the limited number of controlling factors in the models (Hyytia-Trees et al., 2000). They stated that with similar types of products (meat-based sous vide) that rely on refrigeration to inhibit the growth of non-proteolytic C. botulinum, predictive models should not be used and that the safety evaluation be based on challenge studies.
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