On the Time Presentation in Differential Rate Equations of Dynamic Microbial Inactivation and Growth

Abstract

A dynamic (e.g., non-isothermal) kinetic model of microbial survival during a lethal process or growth under favorable conditions is either in the form of a differential rate equation from the start or obtained from an algebraic static model by derivation. Examples of the first kind are the original and modified versions of the logistic (Verhulst) equation and of the second the dynamic Weibull survival or Gompertz growth models. In the first-order inactivation kinetics, the isothermal logarithmic survival rate is a function of temperature only. Therefore, converting its static algebraic form into a dynamic differential rate equation, or vice versa, is straightforward. There is also no issue where both the static and dynamic versions of the survival or growth model are already in the form of a differential rate equation as in the logistic equation of growth. In contrast, converting the nonlinear static algebraic Weibull survival model or the Gompertz growth model into a dynamic differential rate equation, requires replacement of the nominal time t by t*, defined as the time which corresponds to the momentary static survival or growth ratio at the momentary temperature. This replacement of the nominal time in the rate equation with a term that contains the momentary survival or growth ratio eliminates inevitable inconsistencies and renders the resulting dynamic model truly predictive. The concept is demonstrated with simulated dynamic microbial survival patterns during a hypothetical thermal sterilization where the temperature fluctuates and with simulated dynamic microbial growth in storage where the temperature oscillates.