The probability of bacterial spores surviving a thermal process: The 12D myth and other issues with its quantitative assessment.

The concepts of "D-value," "thermal death time" and "commercial sterility," innovative and useful at their inception, are based on untenable assumptions, notably that the log-linear isothermal inactivation model has universal applicability, that extrapolation over several orders of magnitude below the detection level is permissible, and that total microbial inactivation is theoretically impossible. Almost all commonly observed inactivation patterns, the log-linear is just a special case, can be described by both deterministic and fully stochastic models, examples of which are given. Unlike the deterministic, the stochastic models predict either complete elimination of the targeted cells or spores in realistic finite time, or residual survival. In most cases, the published survival data do not contain enough information to establish which actually happens. The microbial safety of thermally processed foods can be compromised not only by under-processing but also by a variety of mishaps whose occurrence probabilities are unrelated to the inactivation kinetics. Moreover, the available sampling plans to detect microbial contamination in sterilized containers through incubation alone are insensitive to levels of potential safety concerns. In principle, many of these issues could be resolved by developing new dramatically improved detection methods and/or verifiable methods to predict very low levels of microbial survival.

Deterministic and stochastic survival models of injured ozonated Giardia cysts.

Giardia cysts exposed to short sublethal ozonation in lake waters continue to die-off well after the ozone complete dissipation. This delayed inactivation can be the manifestation of injured cysts' mortality, which the traditional Chick-Watson-Hom type models of disinfection do not account for. But it can be described by a slightly modified version of a general microbial survival model adapted for injured cysts or other targeted microorganisms surviving disinfection. The downward concavity of the cysts' semi-logarithmic survival ratio vs. time relationships suggests that the cysts' deaths had unimodal temporal distribution. Indeed, the cumulative (CDF) forms of the Weibull and lognormal distribution functions both had excellent fit to the experimental survival data. Such a survival pattern can also be described by a fully probabilistic model devised from the injured cysts' Markov chain, where the mortality's probability rate rises linearly with time. The stochastic model explains the ubiquitous observation that microbial survival curves become increasingly irregular and irreproducible as the number of survivors dwindles, regardless of their concavity degree and direction. Although based on ozonated Giardia cyst data, the concept should be applicable to the delayed mortality of other microorganisms surviving sublethal treatments of other kinds but unable to recover and/or multiply. KEY POINTS: • Deterministic and stochastic survival models can describe delayed inactivation. • The Weibull and lognormal distributions can describe cysts' times to mortality. • Stochastic model explains the progressively growing scatter in survival curves.

Models of the water activity effect on microbial growth rate and initiation.

The role water activity, aw, plays in microbial growth by itself or in conjunction with other factors, notably temperature and pH, has been described mathematically by different algebraic models obtained by fitting experimental growth rate vs. aw relationships. Many of these models have one, two, or all three cardinal parameters, namely the minimal, optimal, and maximal aw, in their formulation. Although they all have good fit as judged by statistical criteria, their different mathematical structures have different ramifications concerning the threshold aw for growth initiation, and the growth pattern around and beyond the optimal aw level where it exists. The focus of this review is on the biological implications of the different growth rate vs. aw models inferred exclusively from their mathematical properties, leaving out any statistical fit considerations. It also describes a recently proposed single-parameter model of monotonic or the monotonic part of experimental growth rate vs. aw curves, which can be combined with a decay term to produce a general conceptual model of peaked and monotonic microbial growth rate vs. aw relationships over the entire aw range. KEY POINTS: • Traditional and new growth rate vs. aw models are presented and their implications compared. • Analogy between aw and the temperature or pH effect on microbial growth rate is reassessed. • Cardinal parameters alone do not establish a unique growth rate vs. aw relationship.

Modeling the dynamic kinetics of microbial disinfection with dissipating chemical agents-a theoretical investigation.

The most notable microbial survival models of disinfection kinetics are the original and modified versions of the static Chick-Watson-Hom's (CWH) initially developed for water chlorination. They can all be viewed as special cases of the Weibull survival model, where the observed static curve is the cumulative form (CDF) of the times at which the individual targeted microbes succumb to the treatment. The CWH model time's exponent is the distribution's shape factor, and its concentration-dependent rate parameter represents the distribution's scale factor's reciprocal. Theoretically, the concentration- dependence of the Weibull model's rate parameter need not to be always in a form of a power-law relationship as the CWH model requires, and two possible alternatives are presented. Apart from being chemically reactive, most chemical disinfectants are also volatile, and their effective concentration rarely remains constant. However, the published dynamic versions of the original CWH model are mathematically incongruent with their static versions. The issue is nonexistent in the dynamic version of the Weibull or other distribution-based models, provided that the momentary inactivation rate is expressed as the static rate at the momentary concentration, at the time that corresponds to the momentary survival ratio. The resulting model is an ordinary differential equation (ODE) whose numerical solution can describe survival curves under realistic regular and irregular disinfectant dissipation patterns, as well as during the disinfectant dispersion and/or its replenishment. KEY POINTS: • The Chick-Watson-Home models are treated as special cases of the Weibull distribution. • Dynamic microbial survival curve described as ordinary differential equation solution. • Survival rate models of disinfectant dissipation and replenishment patterns presented.

Kinetic parameters of thiamine degradation in NASA spaceflight foods determined by the endpoints method for long-term storage.

Retention of labile vitamins such as thiamine (vitamin B1) in NASA spaceflight foods intended for extended-duration missions is critical for the health of the crew. In this study, the degradation kinetics of thiamine in three NASA spaceflight foods (brown rice, split pea soup, BBQ beef brisket) during storage was determined for the first time, using an interactive isothermal model developed by our group. Results showed that brown rice and split pea soup demonstrated resistance to thiamine degradation, while thiamine in beef brisket was less stable. Model-predicted thiamine retention in brown rice stored at 20 °C for 720 days was 55% of the original thiamine content after thermal processing, 42% for split pea soup, and 3% for beef brisket. Water activity, moisture content, and pH differences did not sufficiently explain the variation in the degradation kinetics of thiamine among these foods.

The instrumental texture profile analysis revisited.

Although innovative at the time of their inception, all the historic and extant instrumental texture profile analysis (TPA) versions have serious methodological flaws. Their measured and calculated parameters, for example, "hardness," "brittleness," and "cohesiveness," bear only a remote relationship to the same properties as understood in material science and other disciplines. The TPA parameters are supposedly objective measures of the tested food's textural attributes. But because they are all specimen size-dependent, they cannot be considered intensive material properties. Also, because the arbitrary test conditions, notably the specimen and probe's geometries and the set deformation level significantly affect the TPA parameters' magnitudes, assigning them textural term leads to logical inconsistencies, making their relationship to the food's actual properties even more difficult to establish. It is doubtful that the instrumental TPA parameters indeed describe the same properties in different foods and sometimes even within the same food, as in ripening juicy fruits and certain soft cheeses. It is proposed that the TPA parameters currently in use be replaced by a list of mechanical and other physical properties determined by testing methods recognized by material scientists, such as "yield stress," "strain at failure," "stiffness," and "toughness," perhaps supplemented by a quantitative measure of "juiciness" and/or the acoustic signature's features, especially developed for the particular food. It is also proposed that instead of correlating such intensive material properties with sensory evaluations described by a predetermined sensory vocabulary, they should be used to study the distribution or spectrum of humans' verbal responses, expressed in their own chosen terms.

Modeling the degradation kinetics of ascorbic acid.

Most published reports on ascorbic acid (AA) degradation during food storage and heat preservation suggest that it follows first-order kinetics. Deviations from this pattern include Weibullian decay, and exponential drop approaching finite nonzero retention. Almost invariably, the degradation rate constant's temperature-dependence followed the Arrhenius equation, and hence the simpler exponential model too. A formula and freely downloadable interactive Wolfram Demonstration to convert the Arrhenius model's energy of activation, Ea, to the exponential model's c parameter, or vice versa, are provided. The AA's isothermal and non-isothermal degradation can be simulated with freely downloadable interactive Wolfram Demonstrations in which the model's parameters can be entered and modified by moving sliders on the screen. Where the degradation is known a priori to follow first or other fixed order kinetics, one can use the endpoints method, and in principle the successive points method too, to estimate the reaction's kinetic parameters from considerably fewer AA concentration determinations than in the traditional manner. Freeware to do the calculations by either method has been recently made available on the Internet. Once obtained in this way, the kinetic parameters can be used to reconstruct the entire degradation curves and predict those at different temperature profiles, isothermal or dynamic. Comparison of the predicted concentration ratios with experimental ones offers a way to validate or refute the kinetic model and the assumptions on which it is based.

Temperature-viscosity models reassessed.

The temperature effect on viscosity of liquid and semi-liquid foods has been traditionally described by the Arrhenius equation, a few other mathematical models, and more recently by the WLF and VTF (or VFT) equations. The essence of the Arrhenius equation is that the viscosity is proportional to the absolute temperature's reciprocal and governed by a single parameter, namely, the energy of activation. However, if the absolute temperature in K in the Arrhenius equation is replaced by T + b where both T and the adjustable b are in °C, the result is a two-parameter model, which has superior fit to experimental viscosity-temperature data. This modified version of the Arrhenius equation is also mathematically equal to the WLF and VTF equations, which are known to be equal to each other. Thus, despite their dissimilar appearances all three equations are essentially the same model, and when used to fit experimental temperature-viscosity data render exactly the same very high regression coefficient. It is shown that three new hybrid two-parameter mathematical models, whose formulation bears little resemblance to any of the conventional models, can also have excellent fit with r2 ∼ 1. This is demonstrated by comparing the various models' regression coefficients to published viscosity-temperature relationships of 40% sucrose solution, soybean oil, and 70°Bx pear juice concentrate at different temperature ranges. Also compared are reconstructed temperature-viscosity curves using parameters calculated directly from 2 or 3 data points and fitted curves obtained by nonlinear regression using a larger number of experimental viscosity measurements.

Theoretical study of aerobic vitamin C loss kinetics during commercial heat preservation and storage.

The non-isothermal aerobic degradation of ascorbic acid (AA), and the formation and subsequent degradation of the dehydroascorbic acid (DHAA) are described by two simultaneous rate equations, assuming that all three underlying reactions follow first or other fixed order kinetics. Also assumed is that the temperature-dependence of these three reactions' rate-constants follows the exponential model, a simpler substitute for the traditional Arrhenius equation. The effective momentary vitamin C's concentration is assumed to be the sum of the AA's and DHAA's momentary concentrations. Plots of the rate equations' numerical solutions for non-isothermal temperature histories, such as those that exist in actual heat preservation processes and commercial storage of foods, allow examining the temperature regime's role in the vitamin C's retention, at least qualitatively, and to evaluate its loss pattern when DHAA is initially absent or already present in the food. The temperature profile can be expressed algebraically or as an interpolating function constructed for digitized data. With the generalized model's version, characterization of the vitamin loss kinetics may require at least nine independent parameters, which ought to be known a priori, assumed on the basis of literature reports or determined experimentally. However, for many practical applications the number of kinetic parameters can be reduced to six. Isothermal loss is a special case, which can be used to test the calculation procedure by comparing the numerical solutions for quasi-isothermal temperature histories with the analytical solutions for truly isothermal temperature profiles. A version of the calculation and simulation procedure has been posted on the Internet as a freely downloadable interactive Wolfram Demonstration, which can be used as a tool to examine the effect of actual and hypothetical temperature histories on the vitamin's retention during heat processing and storage. It can also be used to assess the impact of deviations from the assumed kinetic order and variability in the model's kinetic parameters on the vitamin's retention pattern.

A New Look at Kinetics in Relation to Food Storage.

Modern mathematical software and user-friendly interactive programs can simplify and speed up kinetics calculations. They also open the way for new approaches to storage data gathering and analysis. This is demonstrated with a recently introduced simple exponential model that is interchangeable with the Arrhenius equation and endpoints and successive points methods and that estimates chemical degradation kinetics parameters from a small number of isothermal or nonisothermal experimental data. Also presented are a method to determine shelf life using two chemical markers and a global phenomenological model for peaked reactions, such as those encountered in lipid oxidation. Also recently introduced are freely downloadable Wolfram Demonstrations and other interactive software to generate, visualize, examine, and/or compare actual or hypothetical storage scenarios in minutes. They include programs that solve pairs of simultaneous nonlinear algebraic or differential rate equations by passing two reconstructed degradation curves, or a single nonisothermal curve, through two entered experimental points by moving the degradation parameters' sliders on the screen.

Crunchiness Loss and Moisture Toughening in Puffed Cereals and Snacks.

Upon moisture uptake, dry cellular cereals and snacks loose their brittleness and become soggy. This familiar phenomenon is manifested in smoothing their compressive force-displacement curves. These curves' degree of jaggedness, expressed by their apparent fractal dimension, can serve as an instrumental measure of the particles' crunchiness. The relationship between the apparent fractal dimension and moisture content or water activity has a characteristic sigmoid shape. The relationship between the sensorily perceived crunchiness and moisture also has a sigmoid shape whose inflection point lies at about the same location. The transition between the brittle and soggy states, however, appears sharper in the apparent fractal dimension compared with moisture plot. Less familiar is the observation that at moderate levels of moisture content, while the particles' crunchiness is being lost, their stiffness actually rises, a phenomenon that can be dubbed "moisture toughening." We show this phenomenon in commercial Peanut Butter Crunch® (sweet starch-based cereal), Cheese Balls (salty starch-based snack), and Pork Rind also known as "Chicharon" (salty deep-fried pork skin), 3 crunchy foods that have very different chemical composition. We also show that in the first 2 foods, moisture toughening was perceived sensorily as increased "hardness." We have concluded that the partial plasticization, which caused the brittleness loss, also inhibited failure propagation, which allowed the solid matrix to sustain higher stresses. This can explain other published reports of the phenomenon in different foods and model systems.

Predicting anthocyanins' isothermal and non-isothermal degradation with the endpoints method.

The thermal degradation of anthocyanins in a variety of media and over a large temperature range is known to follow first-order kinetics, and the temperature-dependence of the exponential rate constant a two-parameter model. These parameters can be estimated from the initial and final concentrations of only two isothermal or non-isothermal heat treatments by numerically solving a pair of simultaneous equations of which they are the two unknowns. Once calculated they can be used to reconstruct the entire degradation curves and predict those of other heat treatments in a pertinent temperature range. Commercial mathematical software can do the calculations, as demonstrated with computer simulations and published data on the isothermal and non-isothermal degradation of anthocyanins. The endpoints method's predictions were confirmed by comparison to the reported experimentally determined final concentrations. Where applicable, the method will eliminate the need to record sets of whole isothermal degradation curves in studies of the kinetics of anthocyanins' degradation.

Simulating shelf life determination by two simultaneous criteria.

The shelf life of food and pharmaceutical products is frequently determined by a marker's concentration or quality index falling below or surpassing an assigned threshold level. Naturally, different chosen markers would indicate different shelf life for the same storage temperature history. We demonstrate that if there are two markers, such as two labile vitamins, the order in which their concentrations cross their respective thresholds may depend not only on their degradation kinetic parameters but also on the particular storage temperature profile, be it isothermal or non-isothermal. Thus, at least theoretically, the order observed in accelerated storage need not be always indicative of the actual order at colder temperatures, except where the two degradation reactions follow the same kinetic order and their temperature-dependence rate parameter is also the same. This is shown with simulated hypothetical degradation reactions that follow first or zero order kinetics and whose rate constant's temperature-dependence obeys the exponential model. It is also demonstrated with simulated hypothetical Maillard reaction's products whose synthesis rather than their degradation follows pseudo zero order kinetics. The software developed to do the simulations and calculate the thresholds crossing points has been posted on the Internet as a freely downloadable interactive Wolfram Demonstration, which can be used as a tool in storage studies and shelf life prediction. In principle, the methodology can be extended from two to any number of markers.

Predicting chemical degradation during storage from two successive concentration ratios: Theoretical investigation.

When a vitamin's, pigment's or other food component's chemical degradation follows a known fixed order kinetics, and its rate constant's temperature-dependence follows a two parameter model, then, at least theoretically, it is possible to extract these two parameters from two successive experimental concentration ratios determined during the food's non-isothermal storage. This requires numerical solution of two simultaneous equations, themselves the numerical solutions of two differential rate equations, with a program especially developed for the purpose. Once calculated, these parameters can be used to reconstruct the entire degradation curve for the particular temperature history and predict the degradation curves for other temperature histories. The concept and computation method were tested with simulated degradation under rising and/or falling oscillating temperature conditions, employing the exponential model to characterize the rate constant's temperature-dependence. In computer simulations, the method's predictions were robust against minor errors in the two concentration ratios. The program to do the calculations was posted as freeware on the Internet. The temperature profile can be entered as an algebraic expression that can include 'If' statements, or as an imported digitized time-temperature data file, to be converted into an Interpolating Function by the program. The numerical solution of the two simultaneous equations requires close initial guesses of the exponential model's parameters. Programs were devised to obtain these initial values by matching the two experimental concentration ratios with a generated degradation curve whose parameters can be varied manually with sliders on the screen. These programs too were made available as freeware on the Internet and were tested with published data on vitamin A.

Antimicrobial N-halamine modified polyethylene: characterization, biocidal efficacy, regeneration, and stability.

UNLABELLED: Development of antimicrobial materials that regenerate antimicrobial activity represents a novel technology in preventing microbial cross-contamination. We report a method for the application of regenerably antimicrobial N-halamines onto the surface of polyethylene (PE) materials through layer-by-layer (LbL) assembly of polyethyleneimine and poly(acrylic acid). A total of 5, 10, 15, and 20 bilayers were applied. Modified PE had from 49.3 to 293.5 nmol cm(-2) antimicrobial N-halamines from 5 to 20 bilayers after 10 min of chlorination. Each variant of N-halamine modified PE was able to reduce by >5 logarithmic cycles Listeria monocytogenes. The stability of N-halamine modified PE was characterized after extended exposure to chlorine, acidic solutions, and an alkaline cleaner. After an initial conditioning period, materials generated more than double the quantity of N-halamines present on as prepared materials, retaining regenerability for up to 100 chlorination cycles. After the equivalent of 300 washing cycles by buffers (pH values 3, 5, and 7) or a commercial alkaline detergent, there was no change in the number of antimicrobial N-halamines on the modified materials. These results indicate that the reported LbL deposition technique results in antimicrobial N-halamine materials capable of long-term reuse and exposure to harsh chemicals as expected in a food-processing environment. Such robust, regenerably antimicrobial materials could be an effective technology in the food industry to prevent cross-contamination of pathogenic and spoilage microorganisms. PRACTICAL APPLICATION: The food contact surface of polyethylene was modified by layer-by-layer deposition of 2 polymers, resulting in a rechargeably antimicrobial surface. Repeated exposure to chlorine regenerated its antimicrobial activity, resulting in greater than 99.999% reduction in Listeria monocytogenes. Materials were stable against repeated washing and exposure to acidic environments. These food contact materials could support current cleaning and sanitization protocols in improving food safety in the processing environment.

Expanded Fermi Solution to estimate health risks or benefits from interacting factors.

Food related health risk/benefit factors might be additive or multiplicative, but some can be interrelated in ways that are neither or both. Since there is always inherent uncertainty concerning the magnitude of risk/benefit factors their contribution is better represented by a probable range than a single numerical value. With the Expanded Fermi Solution method, random values of the factors are generated within their respective ranges and used to calculate the combined risk or benefit based on the chosen mathematical model. In the case of additive or multiplicative independent factors, the combined risks/benefits so calculated have approximately normal or lognormal distribution, respectively, as anticipated from the Central Limit Theorem. The distribution's mode, i.e., the most frequent value, is considered the risk's best estimate. In interactive factors, the emergence of a specific parametric distribution is not guaranteed, but the histogram of the randomly generated combined risks or benefits can be used to identify the best estimate. This is demonstrated with three abstract interactive risk/benefit models of increasing number of parameters and complexity, whose calculation has been automated and posted on the Internet as freely downloadable interactive Wolfram Demonstration. Also given is a hypothetical but realistic example of dose-response based microbial risk assessment where uncertain bacterial growth parameters are involved, which can be implemented with a new Wolfram Demonstration. The methodology and software allow rapid examination of numerous combinations of interactive factors and evaluation of their potential effect on a food's or supplement's risk or benefit.

Modeling of fungal and bacterial spore germination under static and dynamic conditions.

Isothermal germination curves, sigmoid and nonsigmoid, can be described by a variety of models reminiscent of growth models. Two of these, which are consistent with the percent of germinated spores being initially zero, were selected: one, Weibullian (or "stretched exponential"), for more or less symmetric curves, and the other, introduced by Dantigny's group, for asymmetric curves (P. Dantigny, S. P.-M. Nanguy, D. Judet-Correia, and M. Bensoussan, Int. J. Food Microbiol. 146:176-181, 2011). These static models were converted into differential rate models to simulate dynamic germination patterns, which passed a test for consistency. In principle, these and similar models, if validated experimentally, could be used to predict dynamic germination from isothermal data. The procedures to generate both isothermal and dynamic germination curves have been automated and posted as freeware on the Internet in the form of interactive Wolfram demonstrations. A fully stochastic model of individual and small groups of spores, developed in parallel, shows that when the germination probability is constant from the start, the germination curve is nonsigmoid. It becomes sigmoid if the probability monotonically rises from zero. If the probability rate function rises and then falls, the germination reaches an asymptotic level determined by the peak's location and height. As the number of individual spores rises, the germination curve of their assemblies becomes smoother. It also becomes more deterministic and can be described by the empirical phenomenological models.

The Arrhenius equation revisited.

The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time. The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the model's validity. Consequently, the slope of the plot has been used to calculate the reaction or processes' "energy of activation," usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(T(reference))] = c(T-T(reference)). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a "mole" cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the model's equation rather than to the existence of a temperature independent "energy of activation." If T+273.16°C in the Arrhenius model's equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and T(reference), the plot of Ln k(T) vs. 1/(T+b) will always appear almost perfectly linear. Both the modified Arrhenius model version having the arbitrary constant b, Ln[k(T)/k(T(reference)) = a[1/ (T(reference)+b)-1/ (T+b)], and the exponential model can faithfully describe temperature dependencies traditionally described by the Arrhenius equation without the assumption of a temperature independent "energy of activation." This is demonstrated mathematically and with computer simulations, and with reprocessed classical kinetic data and published food results.

On quantifying nonthermal effects on the lethality of pressure-assisted heat preservation processes.

Direct experimental identification and quantification of the pressure contribution to a pressure-assisted sterilization process efficacy is difficult. However, dynamic kinetic models of thermal inactivation can be used to assess the lethality of a purely thermal process having the same temperature profile. Thus, a pressure-assisted process' temperature record can be used to generate a corresponding purely thermal survival curve with parameters determined in conventional heating experiments. Comparison of the actual final survival ratio with that calculated for the purely thermal process would reveal whether the hydrostatic pressure had synergistic or antagonistic effect on bacterial spores survival. The effect would be manifested in the number of log cycles subtracted or added to the survival ratio, and in the length of time at the holding temperature needed to produce the final survival ratio of the combined process. A set of combined treatments would reveal how the temperature and pressure profiles affect the pressure's influence on the process' lethality to either vegetative cells or spores. The need to withdraw samples during the thermal and combined processes would be avoided if the thermal survival parameters could be calculated by the "three endpoints method," which does not require the entire survival curve determination. Currently however, this method is limited to thermal inactivation patterns characterized by up to 3 survival parameters, the Weibull-Log logistic (WeLL) model, for example.