These models are also referred as growth/no growth (G/NG) interface models or G/NG boundary models, or "growth boundary'' models or simply "growth limits'' models. Traditional growth modeling (primary and secondary models) approach is suitable for determining the growth of most foodborne pathogens and spoilage microorganisms. When there are regulatory limits for certain pathogens in particular foods, G/NG boundary models are the most appropriate models. For example, G/NG models are more suitable for L. monocytogenes for which there is a zero tolerance in the USA.

The growth or no growth conditions for several combinations of various environmental conditions such as temperature, pH, aw, and salt content have to be determined. At extreme growth ranges, the growth of microorganisms is very erratic and makes the development of these models more complex. Thus, the probability of growth is modeled due to uncertainties involved in bacterial growth under sub-optimal conditions. It is important to note that the probability of growth is strongly time dependent. Therefore, these models have to be defined for certain time period, which is typically the storage life of the product.

Logistic models are commonly used to describe the probability of growth for given conditions. Logistic models are defined as (Ratkowsky, 2002)

log (r—p) = logit[p] = b0 + b\XT + b2XPH + b3 Xaw + b4XNO2

ebo + bl xt + b2XpH + bxaw + b^XNO2 1 eb0 + b1XT + b2XpH + b3Xaw + b4XNO2

where Xt = ln (T - Tmin), XpH = ln (pH - pHmin), Xaw = ln (aw - awmin), XNo2 = ln(NO2 - max -NO2) are environmental variables such as temperature, pH, aw, or salt content and P is the probability of observing growth (a value of 1).

The logit(P) can be interpreted as log of ratio of odds for growth and odds for no growth.

^ / P \ ^ / probability of growth \ ^ / probability of growth \ \1 - PJ \1 -probability of growth) \1 -probability of no growth J

= log(odds ratio)

If the probability of growth (P) is 0.75, then the probability of no growth is 0.25. Then, the odds of growth are 3 (0.75/0.25). Thus, there is three times more probability to observe growth than no growth. The log (odds) is then modeled as a function of logarithm of environmental variables. One of the major challenges in developing G/NG boundary models is the large number of data points necessary, which requires significant time and resources to generate the data. Marc, Pin, and Baranyi (2005) used the ComBase database to collect large amount of growth/no growth data for several pathogens and developed G/NG models. Figure 22.1 shows the G/NG model for L. monocytogenes. A contour boundary can be drawn for each probability level of growth in a multidimensional environmental space.

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