Dynamic Model of Heat Inactivation Kinetics for Bacterial Adaptation

The Weibullian-log logistic (WeLL) inactivation model was modified to account for heat adaptation by introducing a logistic adaptation factor, which rendered its “rate parameter” a function of both temperature and heating rate. The resulting model is consistent with the observation that adaptation is primarily noticeable in slow heat processes in which the cells are exposed to sublethal temperatures for a sufficiently long time. Dynamic survival patterns generated with the proposed model were in general agreement with those of Escherichia coli and Listeria monocytogenes as reported in the literature. Although the modified model''s rate equation has a cumbersome appearance, especially for thermal processes having a variable heating rate, it can be solved numerically with commercial mathematical software. The dynamic model has five survival/adaptation parameters whose determination will require a large experimental database. However, with assumed or estimated parameter values, the model can simulate survival patterns of adapting pathogens in cooked foods that can be used in risk assessment and the establishment of safe preparation conditions.Combined with heat transfer data or models, microbial survival kinetics, especially of bacteria or spores, is extensively used to determine the safety of industrial heat preservation processes like canning, extant or planned. The same is true for milder heat processes such as milk and fruit pasteurization. However, survival models are also a valuable tool to assess the safety of prepared foods, especially those made of raw meats, poultry, and eggs, where surviving pathogens can be a public health issue.The heat resistance of a bacterium, or any other microorganism, is almost always determined from a set of its isothermal survival curves, recorded at several lethal temperatures. The kinetic models, which define the heat resistance parameters, may vary, but the calculation procedure itself is usually the same. First, the experimental isothermal survival data are fitted with what is known as the “primary model.” Once fitted, the temperature dependence of this primary model''s coefficients is described by what is known as the “secondary model.” When combined with a temperature profile expression, T(t), and incorporated into the inactivation rate equation, the result is a “tertiary model,” which enables its user to predict the organism''s survival curve under any static or dynamic (i.e., nonisothermal) conditions.The traditional log-linear (“first-order kinetic”) model is the best-known primary survival model, and it is still widely used in sterility calculations in the food, pharmaceutical, and other industries. Traditionally, it has been assumed that the D value calculated with this model has a log-linear temperature dependence or, alternatively, that the temperature effect on the exponential rate constant, k, the D value''s reciprocal, follows the Arrhenius equation. However, accumulating experimental evidence in recent years indicates that bacterial heat inactivation only rarely follows the first-order kinetics and that there is no reason that it should (3, 18, 29). Nonlinear survival curves can be described by a variety of mathematical models (6). Perhaps the most frequently used in recent years is the Weibullian model, of which the traditional log-linear model is a special case—see below.Regardless of the log-linearity issue, none of the above-mentioned models accounts for adaptation, the ability of certain bacterial cells to adjust their metabolism in response to stress in order to increase their survivability (2, 10, 26, 27, 28). A notable example is Escherichia coli. Its cells can produce “heat shock proteins,” which help them to survive mild heat treatments (1, 11). Other organisms, Salmonella enterica and Bacillus cereus among them, can also develop defensive mechanisms that help them to survive in an acidic environment (8, 9, 13). Whether adaptation allows the cells to avoid injury or to repair damage once it has occurred, or both, should not concern us here. (Injury and recovery, although related, are a separate issue, one which is amply discussed in the literature. Their quantitative aspects and mathematical modeling are discussed elsewhere 5].)The cells'' ability to augment their resistance is not unlimited, and it takes time for the cells to activate the protective system and synthesize its chemical elements (10, 12). Consequently, the effect of heat adaptation on an organism''s survival pattern becomes measurable only at or at slightly above what''s known as the “sublethal” temperature range. Under dynamic conditions, therefore, adaptation can be detected only when the heating rate is sufficiently low to allow the cells to respond metabolically to the heat stress prior to their destruction.Several investigators have reported and discussed the quantitative aspects of adaptation (25, 27, 28). When it occurs, adaptation is noticed as a gap between survival curves determined at low heating rates and those predicted by kinetic models whose parameters had been determined at high lethal temperatures (7, 8, 9, 27, 28). The question is how to modify the inactivation kinetic model so that it can properly account for adaptation at low heating rates while maintaining its predictive ability at high rates and clearly lethal temperatures. Stasiewicz et al. (25) have recently given a partial answer to this question. They started with the Weibullian inactivation model (see below) and assumed that its rate parameter''s temperature dependence follows a modified version of the Arrhenius equation. Using this model and experimental data for Salmonella bacteria, they showed that a “pathway-dependent model” is more reliable than a “state-dependent model.”The objectives of our work were to develop a variant of the Weibullian-log logistic (WeLL) inactivation model to account for dynamic adaptation and to demonstrate its applicability with reported adaptive survival patterns exhibited by Escherichia coli and Listeria monocytogenes, two organisms of food safety concern.