## Model Limitations

A model is a compromise between the situation of possessing sufficient complexity to include all the factors that affect microbial behavior and the need to keep the model simple with factors that can be readily known by the user. The appropriateness and accuracy of a given model may vary with the specific application. Most of the current models were developed in broth cultures. Experience has shown that growth in a food corresponds closely to growth in broth if the broth and food temperature, pH, and salt levels are equivalent.

However, if a food has another factor that limits microbial growth, such as high lactate concentrations or low water activities from other humectants, the model may not be appropriate for making predictions for that food. Models estimate values within the ranges of the factors used in the development of the model. Extrapolating beyond the range of the data may lead to erroneous estimates, especially for the empirical models. Comparing the behavior of a pathogen in a specific food of interest under a few conditions is essential before fully trusting a model's predictions for use in that food.

Usually models are made of cocktails containing three to six bacterial strains. Studies show that different strains of the same pathogen vary greatly in survival and thermal inactivation times and also in growth parameters. The ratio of the standard deviation to the mean for the thermal inactivation D values of 17 strains of Salmonella enteritidis was 0.26 at 57.2°C and 0.28 at 60°C (Shah et al., 1991). This means that to include 95% of the strains, the two-standard deviation range is from ~50% to 150% of the mean. How strains used in the model compare with the possible strains that may be present in a food is usually unknown. With a cocktail, essentially the fastest-growing or longest-surviving organism or strain is modeled, and the modeler hopes the selected cocktail includes a strain representing the fastest growing or hardiest likely to be present in a food. However, models based on cocktails do not provide information on variations between strains. The confidence intervals represent that of the regression equation and modeling process, not the variation that would be encountered between strains likely to be present in a food.

Most models do not consider the influence of the natural spoilage flora on pathogen behavior. Lactic acid-producing flora can reduce the pH, and many microbial species produce bacteriocins that inhibit growth of other species. The extent that the relatively low levels of natural flora on high-quality foods affect the low levels of pathogens that usually occur in a contaminated food is not well understood.

### Deterministic versus Probabilistic Models

The models described above are determinative or point estimate models. They calculate the mean number of microorganisms expected under specified conditions. As the conditions for growth become less favorable, however, the growth rate decreases and the variation about the mean rate increases. In addition, at the extremes of the unfavorable conditions, the likelihood of growth also decreases. If a series of identical tubes are incubated at decreasing temperatures, the tubes at the favorable temperatures will all show growth. At lower temperatures some tubes will not have growth, even after extended incubation times. Eventually, as the temperature decreases toward the minimal growth temperature, only a few tubes in a set will have growth. To fully characterize the expected growth in the low-temperature range or other extreme condition, both a growth rate and a probability of growth model are needed. In addition to the environmental factors, the probability of growth is strongly dependent on the number of cells present. An aliquot containing high numbers of spores would be more likely to have growth eventually than an aliquot with only a few spores. This situation was explored in time-to-turbidity models for C. botulinum (Whiting and Oriente, 1997) and growth-no growth boundary models (Rat-kowsky and Ross, 1995).

REGULATORY, INDUSTRIAL, AND INTERNATIONAL IMPLICATIONS Process Modeling and Risk Assessment

To model a series of processing steps or changing environmental conditions, a food process can be separated into a series of unit operations and the appropriate model can be used for each step. A deterministic process model for a frozen ground meat patty was presented by Zwietering and Hastings (1997). The process has 16 individual operations and includes the initial contamination of both the meat and spice mixture. The model provides for rework (defective patties are collected and added back to the beginning of the process) and dead spaces in the equipment where meat can reside for a long period of time and bacterial growth can occur before the meat falls back into the product flow. The model shows the expected microbial population at the end of the process and indicates which steps allow growth. With this information, the food technologist can change the processing parameters, such as microbial quality of the spices or temperature, and estimate the change in microbial numbers at the end of the process. With information on the occurrence of a pathogen in the raw ingredients and designation of the food safety objective (the frequency and level of pathogen determined to be acceptable in the product), the process can be designed to yield an acceptable product. A similar model estimates the increase in Bacillus cereus cells during the production of vacuum-packed cooked potatoes (van Gerwen and Zwietering, 1998).

### Risk Assessment

Microbial risk assessments follow several paradigms of hazard identification, exposure assessment, hazard characterization (or dose-response), and risk characterization (ICMFS, 1998; NACMCF, 1998; Chapter 3). For process modeling of a food system the flow is typically the number and frequency of organisms in the raw ingredients (Marks and Coleman, 1998), linking unit operations with growth, survival or inactivation models, consumption data, and the impact on public health (Coleman and Marks, 1998).

Variation and Uncertainty, Simulation Modeling

The deterministic meat patty and cooked potatoes process models calculate single values for each step in the process with singular input parameter values.

This approach omits the inherent variation and uncertainty in both process inputs and model outputs (Vose, 1998). Variation refers to the real differences that occur about a parameter, for example, different strains of a microorganism have different growth rates and D values. Each strain could be characterized, but it is unknown which strain may be present in a food at a given time; therefore, a single growth rate or D value cannot fully describe what may happen in the future. Likewise, when thermal processes reduce the level of a pathogen to a few or less per package, their occurrence in a particular package is typically dependent on binomial and Poisson distributions. Variation can be reduced by redesign of the process or equipment; better control of the oven temperature would reduce the variation in thermal inactivation of microorganisms in the food.

Uncertainty refers to our lack of knowledge. More precise or extensive measurement and monitoring can reduce this uncertainty. Estimates for the length of time an egg is in a retail store or the degree of Salmonella inactivation during home cooking of an egg would be examples with high uncertainty. In practice, both variation and uncertainty are present in most parameters in microbial models.

Because of variation and uncertainty, each parameter has a distribution of values that it might achieve in any specific instance. This distribution can be described by a variety of functions such as normal, log normal, exponential, beta, or triangle and the appropriate parameter values that describe that distribution, that is, mean and standard deviation. Distributions frequently are skewed, with more of the occurrences toward one end than the other. Distributions also may be described by a frequency graph that simply summarizes experimental data.

Each parameter input value in each unit operation, such as temperature, time, pH of food, and microbial growth rate, has a distribution. Monte Carlo simulation is a computational tool to calculate a model with multiple distributions. The simulation will pick a value for each distribution, calculate each model, and proceed stepwise through the entire process operation (Cassin et al., 1998b). The simulation model repeats the process calculation many times. Each iteration will pick a value from the input distributions. These distributions will tend to cluster about the mean value but will also reflect the range in outcomes likely to occur as a result of the shapes and ranges of the various input distributions. The outputs from the simulations will be distributions instead of single values. A simulation model can indicate which parameters contribute to the absolute value of the output value and which input distributions contribute to the output distributions. Input distributions can easily be changed, the simulations repeated, and the resulting changes in outputs determined.

Several dynamic models for food processes have been reported in the literature. Salmonella enteritidis in pasteurized liquid eggs (Whiting and Buchanan, 1997), shell egg processing, storage, and preparation (FSIS, 1998), E. coli 0157:H7 in ground beef (Cassin et al., 1998a), Salmonella in poultry (Oscar,

1998) and chicken products (Brown et al., 1998), and the presence of L. monocytogenes in cheese made from unpasteurized milk (Bemrah et al., 1998) are the first examples of process models intended to provide understanding of how various parameters interact to affect food safety.

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