## Nonlinear Models 2231 Nonlinearity Issues

The canning industry has enjoyed an enviable record of safety, and thus the concept of logarithmic death of microorganisms has persisted, and is now considered accepted dogma. In spite of this, nonlinear survival curves were reported for some bacteria almost 100 years ago.83 In general there are two classes of nonlinear curves; those with a "shoulder" or lag prior to inactivation, and those that exhibit tailing. These two phenomena may be present together, or with other observed kinetics such as biphasic inactivation. A wide variety of complex inactivation kinetics have been reported, and several of these are shown in Figure 2.6. The theoretical basis for assuming logarithmic behavior for bacteria is based on the assumptions that bacterial populations are homogeneous with respect to thermal tolerance, and that inactivation is due to a single critical site per cell.83 Both of these assumptions have been questioned, and thus concerns have been raised regarding the validity of extrapolation of linear inactivation curves.84,85

Stringer et al.82 have summarized the possible explanations for nonlinear kinetics into two classes: those due to artifacts and limitations in experimental procedure and those due to normal features of the inactivation process. The first class encompasses such limitations as variability in heating procedure; use of mixed cultures or populations; clumping; protective effect of dead cells; method of enumeration; and poor statistical design. The second class includes such situations as possible multiple hit mechanisms; natural distribution of heat sensitivity; and heat adaptation. These two classes roughly parallel the two concepts reviewed by Cerf85 to explain tailing in bacterial survival curves. The first of these (the "mechanistic" approach) also makes the assumption of homogeneity of cell resistance and proposes that thermal destruction follows a process analogous to a chemical reaction. In this approach, deviations from linearity are attributed mainly to artifacts; however, tailing is also related to the mechanism of inactivation or resistance. In the "vitalistic" approach, it is assumed that the cells possess a normal heterogeneity of heat resistance; thus survival curves should be sigmoidal or concave upward.85

Time

FIGURE 2.6 Examples of thermal death curves: (a) lag or shoulder, with either linear (dotted line), power law where p > 1 (broken line), or monophasic logistic (solid line) models; (b) concave with power law where p < 1; (c) biphasic logistic; and (d) sigmoidal.

There has been considerable controversy between the two schools of thought, and the literature is divided on the validity of nonlinear survival curves as representing the true state of the cell population. There is certainly evidence that inconsistencies in experimental protocols or the use of incorrect media can lead to artifacts; however, there is little convincing evidence that clumping of cells or the protective effect of dead cells is consistently responsible for nonlinear survivor curves. The current belief is, notwithstanding some contribution by artifacts, that cells do exhibit heterogeneity in thermal sensitivity, and the majority of modeling approaches now make this assumption. There is also inconsistency in actually defining what is meant by an artifact. If one assumes that an artifact in this context is anything that interferes with obtaining a linear death curve, then many of the situations currently classified as artifacts may be natural behavior of cell populations. This is particularly obvious in the study of spore inactivation where standardized suspensions are difficult to obtain, and much effort has been expended to remove artifacts such as genetic variants. The difficulty in obtaining linear kinetics may be a signal that, in most cases, nonlinearity is the norm.

The current theories of microbial inactivation must be revisited in light of recent improved understanding of the effect of heat on microorganisms. We now know that cells do not exist simply as alive or dead, but may also experience various degrees of injury or sublethal damage, which may give rise to apparent nonlinear survival curves.82 The induction of heat resistance in food-borne pathogens due to expression of heat shock proteins has been extensively documented in recent years, and may contribute to apparent nonlinearity, particularly tailing.828687 Thus it appears important to model the actual conditions or situations experienced by bacteria in foods rather than relying on simplifications. Survival modeling should also include a more complete understanding of the molecular events underpinning microbial resistance to the environment.

It seems likely that heterogeneity within bacterial populations is responsible in most cases for nonlinear survival curves, and most recent attempts to model survival employ distributions. The use of distributions to account for nonlinearity is not new; log normal distributions had been suggested for this purpose as early as 1942.83 Other distributions such as logistic, gamma, and Weibull have also been suggested; Weibull is the favored approach at the moment (see later). There is no complete agreement on the use of distributions,83 and it is clear that this approach cannot adequately account for changes in heat resistance occurring during heating.

Our lack of understanding of the key physiological aspects of microbial inactivation and the complexities of nonlinear behavior suggest that a truly mechanistic model for thermal inactivation will not be developed in the near future. One approach to quanti-tating bacterial survival might be the thermal death point concept common to the canning industry. This approach allows one to define the conditions required to achieve a target log reduction, and makes no statement regarding the kinetics of that destruction. This approach has a number of attractive advantages; however, it would still be influenced by such artifacts as changes in heat resistance of a culture and cell injury.83

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