# Model Development ^{*}

## Introduction

The objective of predictive food microbiology is to mathematically describe the growth or decline of food borne microbes under specific environmental conditions. With this ability to describe comes the capability to predict. Predictive microbiologists assert that a particular microorganism's growth or decline is governed by the environment it experiences. This environment includes both intrinsic factors (pH, a_{W}) and extrinsic factors (temperature, gaseous atmosphere). A large number of factors undoubtedly affect the microorganism; however, in most foods only a few exert most of the control. The effect of a factor is assumed to be independent of whether the microbe is in a broth or food (assuming other relevant factors are equivalent).

Although modeling does not usually reveal unexpected microbial behavior, it does quantify the effects from the interactions between two or more factors and allow interpolation of combinations of factors not explicitly tested. Much of the food microbial literature before the advent of predictive modeling defined the limiting conditions for growth when all other factors were near their optimum. However, in many foods practical control of pathogens depends on a combination of preservative factors, none of the factors at levels capable of inhibiting the microorganisms by themselves. Mathematical models are the best way to make predictions in these circumstances.

## Classification of Models

Several schemes have been proposed to categorize microbial models. In this paper, the initial differentiation will be into growth models and inactivation/survival models. Within each category, models will be described as being primary, secondary or tertiary.

Primary models describe changes in microbial numbers or other microbial responses with time. The model may quantify colony forming units per ml, toxin formation, or substrate levels (which are direct measures of the response), or absorbance or impedance (which are indirect measures of the response). A mathematical equation or function describes the change in a response over time with a characteristic set of parameter values. Examples of primary models are the exponential growth rate, Gompertz function, and first-order thermal inactivation. The parameters may, in turn, be reformulated into derived parameters such as the Gompertz lag time or generation time.

Secondary models describe the responses by the parameters of these primary models to changes in environmental conditions such as temperature, pH, or water activity. Examples of secondary models are the response surface model, Arrhenius relationship and square root model.

Tertiary models are computer software routines that turn the primary and secondary models into "user-friendly" programs for model users in the forms of applications software and expert systems. These programs may calculate microbial responses to changing conditions, compare the effects of different conditions, or contrast the behavior of several microorganisms.

Because the models' regression equations usually use the logarithm of the values to normalize the variances, the upper and lower confidence intervals are not equal when the values are converted into real numbers.

## Limitations of Models

### Statistical Limitations:

An important statistic missing from most current secondary and tertiary models is an estimate of the variation around the calculated value. With marginal conditions for growth, the variation between replicates became large. Transformations were used to homogenize the variances for fitting the models. The logarithm of values for time parameters was frequently closer to being normally distributed than untransformed values.

### Biological Limitations:

It is important the model developer clearly specify directly or through the model what the limits of the model are, ie., what microorganisms, what factors, the ranges of each factor and what combinations of factors will give valid answers. The presence of additional inhibitory factors in a food that were not in the model invalidate the model or require cautious interpretation of the predictions. Currently, growth models do not usually include factors such as anion effects from the acidulent used, phosphates, sorbates, and bacteriocins, and humectants other than sodium chloride. No broth models include competition from other microorganisms. Some models developed with foods include the "normal" spoilage flora, but how this flora may change in species and number with plant or season and the effect upon the modeled microorganism is largely unknown.

Because pathogens grow in most foods, the important question then is whether the pathogens will grow to a significant population before the spoilage flora cause the food to be rejected by the consumer. There is a need for systematic modeling of representative classes of spoilage microorganisms so tertiary models can then plot comparative growth curves for both pathogenic and spoilage organisms. For some pathogens with very low infective or toxic dose, such as Listeria, Yersinia and C. botulinum, the criteria may be growth-no growth and the spoilage flora has little significance unless they alter the environment by lowering the pH or produce a bacteriocin.

## Applications of Microbial Models

It must be stressed that models are valuable tools for making predictions and planning Hazard Analysis, Critical Control Point (HACCP) programs. Particularly at present, as models are evolving from the basic research laboratory to use by industry and regulatory agencies, models should be considered as initial estimators of microbial behavior and guides for evaluating potential problems. Models do not completely replace microbial testing nor the judgment of a trained and experienced microbiologist. Models can provide very useful information for making decisions in the following situations:

Time to growth and survival models can estimate whether there is likely to be a risk in a particular food after a specified time-temperature storage. Growth models can aid in setting a pull date governed by growth of a pathogenic or spoilage microorganism.

Identification of critical steps in the process by the model assists in developing a HACCP program. A critical control point can exist where the model indicates that a certain level of a factor permits or suppresses microbial growth. Quantitative estimates of microbial behavior at different levels of the factors can suggest the allowable ranges for that factor.

The consequences of reformulations on growth or inactivation can be estimated. Models show which factor has the major influence and can identify alternative formulations with similar or enhanced resistance to growth.

The consequences of out-of-process events such as lack of intended salt or inadequate refrigeration can be immediately determined. Decisions to rework, rapidly utilize or scrap a product can be made without waiting for testing.

By generating graphs or estimates of the time to a specified microbial population, models can be educational tools, particularly for non-technical people. The model can dramatically demonstrate the importance of maintaining proper refrigeration temperatures or the benefits of high-quality raw materials with lower initial populations.

Using microbial models potentially saves resources, time and money by reducing much of the laboratory work. This permits the laboratory to utilize its resources in other areas. The model will quickly give the ranges of concern for a factor and thereby guide the design of challenge tests, storage trials and other conventional techniques to assess the probability of pathogen growth. Examining the model's predictions increases the understanding of what governs microbial growth or decline in a particular food and thereby gives the processor greater confidence in his process and product. This knowledge enables the manufacturer to create a more sophisticated and effective HACCP program.

Ultimately, what is desired is a risk assessment. What are the chances of becoming ill from a foodborne pathogen after consuming a specific food? This requires quantitative evaluation of three areas. The first is to identify and enumerate all possible sources of contamination. Both the frequency of occurrence and the numbers of pathogens are needed. The second is to understand the physiology, biochemistry and behavior of the pathogens. How fast will they grow or produce toxins under specified conditions? Finally, it is necessary to characterize the human response to the pathogen. What is the infectious dose? These points must be integrated into a publicly trusted, cost-benefit analysis to determine the steps that minimize risk. Current efforts in microbial modeling are making rapid strides in fulfilling the second point.

## Conclusion

The progress in microbial modeling has been impressive and models are becoming a standard research tool and a valuable aid in evaluating and designing food processes. However, it is not yet possible to rely solely upon models to determine the safety of foods and process systems. Laboratory testing is still necessary to unequivocally determine the propensity for pathogen growth or survival in the food product.

^{*}Excerpted from:

Whiting, R. C. 1995. Microbiological modeling. CRC Critical Reviews in Food Science and Nutrition. 35:467-494.