predictive modeling

Predictive food microbiology is a sub-discipline of food microbiology that uses models (i.e., mathematical equations) to describe the growth, survival, or inactivation of microbes in food systems. Mathematical models refer to a set of basic hypotheses supporting the target (bio-) processes which are to be simulated and are possibly algebraic functions and/or differential equations (Baranyi and Roberts, 1995). Therefore, with predictive microbiology, all the knowledge of microbial responses in different environmental conditions is summarized as mathematical equations. McMeekin et al. (2008), stated that “the model is often a simplified description of relationships between observations of the system (responses) and the factors that are believed to cause the observed responses.”

Based on microbial responses, expressed as change in numbers and stress tolerance, the combinations of intrinsic and extrinsic environmental determinants to which microorganisms may be exposed, are divided into the following major domains: the growth era and the domain including the combinations that allow survival or cause death of microorganisms (Booth, 2002). The conditions that lie between these two domains refer to a zone where microbial responses are uncertain and characterized by the growth/no growth interface (Le Marc et al., 2005). This zone is strongly associated with the so-called cardinal values (T, pH, a_w, etc.) for growth and outlines the bio-kinetic range of microbial proliferation. Such values are species- or even strain-dependent and thus, introduce significant variability in the assessment of the impact of marginal growth conditions on microbial growth, an issue commonly encountered in quantitative microbial risk assessment. To remedy that, models have been proposed which embed the theoretical growth-limiting values for critical hurdles, such as temperature, a_w, pH, CO₂ and preservatives as biological meaningful parameters in the model structure. Notably, a theoretical interface also exists between survival and inactivation separating combinations that cause growth cessation but not cellular death from those that are lethal (McKellar et al., 2002).

Depending on the conceptual modeling approach applied to the target biochemical process and the final algebraic form, the models can be characterized as empirical or phenomenological, which mathematically describe specific behavior, and mechanistic or theoretical models with a biological basis, which search for the underlying mechanisms driving already observed phenomena. Polynomial equations are the most common empirical models. These models are easy to use, straightforward and no knowledge of a particular process is required. However, polynomial models have no theoretical foundation and have numerous parameters without biological meaning. Therefore, they do not offer any knowledge to mechanisms underlying a process. Polynomial models are commonly represented as quadratic response surfaces describing the environment dependence of a parameter of a bacterial population (Gibson et al., 1988).

Based on the type of dependent variable that is predicted, the models can be classified as kinetic or probabilistic. Kinetic models predict the extent and rate of growth or inactivation of a microorganism. The growth rate of a microorganism can be modeled in order to be used for making predictions based on the exponential growth of the corresponding microbial population. Kinetic models can be used to predict changes in microbial numbers with time, even if a controlling variable, which can affect growth, is changing (McDonald and Sun, 1999). This type of models constitutes a fundamental model category in predictive microbiology, especially for ready-to-eat foods, since they may assess the exposure of consumers to the doses (levels) of pathogenic bacteria at the time of consumption. The purpose of kinetic models is to estimate the time required for a specified growth or inactivation response to occur under certain intrinsic or extrinsic conditions. Such conditions include temperature, pH, a_w, packaging atmosphere (e.g., CO₂ levels), redox potential (Eh), the rheological properties of the food (structure-associated variables), relative humidity, nutrient content and the concentration of antimicrobial compounds (Theys et al., 2009b; Mejlholm et al., 2010; Møller et al., 2013). Thermal inactivation was the first microbial inactivation process modeled since 1920 by the canned food industry, in order to control the risk of Clostridium botulinum toxigenesis. First-order inactivation models were used to describe a log-linear trend of C. botulinum spores in low acid canned food. Through the slope of inactivation curves the thermal death time was estimated and particularly in low acid canned foods, a 12-decimal reduction (12D) of C. botulinum spores was shown to require exposure to 121°C/15 psi for 15 min. Over the last decades the microbial inactivation modeling was expanded to account for non-thermal inactivation (Buchanan et al., 1997). In addition to the classical linear inactivation curve, the concept of biphasic death, associated with the pre-existence or emergence of a resistant sub-populations throughout exposure to lethal conditions was modeled with non-linear models (Whiting, 1993; Geeraerd et al., 2005). Probabilistic models constitute the corner stone of predicting microbial dynamics, acting as the filter, i.e., likelihood-based decision of the primary microbial response (growth or inactivation) and guiding the selection of the subsequent kinetic modeling tool, i.e., growth or inactivation model, for predicting the change in microbial numbers in time. As such, the fate of microbial populations in foods is eventually dependent on the probability of growth or inactivation phenomena defined by the intrinsic and extrinsic factors of foods and processing environment. From a closer perspective, the behavior of an isogenic (homogeneous) population is the cumulative and stochastic outcome of its individual cells (microscopic level; Kutalik et al., 2005). Each cell within a microbial population is characterized by a variable probability for growth initiation (Koutsoumanis, 2008), followed by a stochastically defined lag time, i.e., sampled from a probability distribution (Francois et al., 2005, 2006a, 2007; Guillier et al., 2005, 2006), both resulting in a fractional growth of the total population with various sub-populations (mesoscopic level) initiating growth on different times (McKellar and Knight, 2000; McKellar, 2001). It has been suggested that under given conditions the geometric lag, i.e., the intersection of the slope at exponential phase with horizontal asymptote at the initial population level, is essentially dependent on the cumulative behavior of the fraction(s) of the initial population, which either possesses the shortest lag time (i.e., the earliest growth starters), and/or the fastest generation time (McKellar and Knight, 2000; Koutsoumanis, 2008). As a mirror image, under lethal conditions, e.g., pH < 3.0, or T > 60°C, the inactivation curve of a microbial population, represented by a curve of survivors (%) vs. time, is the result of the cumulative distribution of the individual cell death time, i.e., the time required to kill every single cell (Aspridou and Koutsoumanis, 2014). In explicit terms, probability models can be used to predict the likelihood of the occurrence of a microbial response as a function of intrinsic and extrinsic factors of foods and processing environment (Ross and Dalgaard, 2004). Microbial responses which have been modeled with this approach include spore germination, toxin formation by C. botulinum, growth initiation and survival or death of bacteria as a result of lethal pH and organic acid combinations. In the context of industrial practice, such models together with cardinal growth models may be of great assistance to HACCP, by offering science-based numerical evidence for setting critical limits, establishing process or product criteria and assessing the compliance of a given process to these limits or the legislative microbiological criteria (e.g., EC Regulation 2073/2005).

All the above model types may be further divided into the following categories, based on the combination of dependent (predicted) and independent (explanatory) variable (Whiting and Buchanan, 1993; McDonald and Sun, 1999):

 i. the primary models, which are used to describe the changes of the microbial population density as a function of time using a limited number of kinetic parameters (e.g., lag time, growth or inactivation rate and maximum population density);

 ii. the secondary models expressing the effect of environmental variables (e.g., temperature, NaCl, pH, etc.) on the kinetic parameters estimated by the primary models;

iii. the tertiary models, which are computer tools that integrate the primary and secondary models into user-friendly units. The wider use of models in the food industry and research depends on the availability of the user-friendly software GroPIN, which encompass predictive models and allow different users to retrieve information from them in a rapid and convenient way (McMeekin et al., 2008, 2013).

The impact on microbial growth of the aforementioned intrinsic and extrinsic variables described by the models is strongly dependent on the structure of food or the model substrate. Based on that, in the following lines, a review is performed of existing modeling approaches accounting for different forms of microbial growth on surfaces, or in the interior of food matrices, either in suspension or immobilized in colonies.

 

Source: Skandamis, P., Jeanson, S., 2015. Colonial vs. planktonic type of growth: mathematical modeling of microbial dynamics on surfaces and in liquid, semi-liquid and solid foods. Front. Microbiol, 29 October 2015, doi: 10.3389/fmicb.2015.01178