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.

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.

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.

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.

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.

A study of the randomly fluctuating microbial counts in foods and water using the Expanded Fermi Solution as a model.

Randomly fluctuating industrial microbial count records, with and without zero counts, were simulated with a version of the Expanded Fermi Solution, originally developed for risk assessment. The basic assumption has been that each individual count is determined by the multiplicative effect of several random factors, which augment or suppress the microbial population size, and in the case of sporadic pathogens, determine the probability of their initial presence too. Records were generated by a series of Monte Carlo simulations in which the factors were specified by ranges and their values chosen randomly within them. The process has been automated and posted as a freely downloadable Wolfram Demonstration on the Internet. The program allows the user to enter and alter the series length, parameters' ranges, and count level deemed dangerous with sliders on the screen. The display includes the chosen factors' ranges, the corresponding generated count record and its histogram, and an estimate of the risk of surpassing the dangerous threshold. Where the record contains no zero counts, the histogram is accompanied by the lognormal distribution, which naturally emerges from the fluctuations' mathematical model. Once the factors are identified and their ranges specified, the method could be used as a tool to analyze, compare, and quantify microbial risks in foods and water.

Construction of food and water borne pathogens' dose-response curves using the expanded Fermi Solution.

Theoretically, the relationship between the number of pathogens that cause acute infection if settling in the gut, N, and that initially ingested, M, can be constructed from the survival probabilities at the different "stations" along the digestive tract. These probabilities are rarely known exactly, but their ranges can be estimated. If for a given N one generates estimates of M using random probabilities within these ranges, the estimates' distribution will be approximately lognormal and its cumulative (CDF) form will represent the pathogen's dose-response curve. The distribution's logarithmic mean and standard deviation can be calculated from the ranges with a formula and used to plot the curve. The method was used to generate dose-response curves of hypothetical food and waterborne pathogens and calculate their infective dose (ID) at 5%, 50%, and 95% probability. The curves were compatible with the Beta Poisson model and robust against minor perturbations in the underlying probabilities' ranges. The calculation and plotting procedure was automated and posted on the Internet as a freely downloadable interactive Wolfram Demonstration. It allows the user to generate, modify, examine, and compare dose-response curves, and to calculate their characteristics, by moving sliders on the screen.

Expanded Fermi solution for estimating the survival of ingested pathogenic and probiotic microbial cells and spores.

The expanded Fermi solution was originally developed for estimating the number of food-poisoning victims when information concerning the circumstances of exposure is scarce. The method has been modified for estimating the initial number of pathogenic or probiotic cells or spores so that enough of them will survive the food preparation and digestive tract's obstacles to reach or colonize the gut in sufficient numbers to have an effect. The method is based on identifying the relevant obstacles and assigning each a survival probability range. The assumed number of needed survivors is also specified as a range. The initial number is then estimated to be the ratio of the number of survivors to the product of the survival probabilities. Assuming that the values of the number of survivors and the survival probabilities are uniformly distributed over their respective ranges, the sought initial number is construed as a random variable with a probability distribution whose parameters are explicitly determined by the individual factors' ranges. The distribution of the initial number is often approximately lognormal, and its mode is taken to be the best estimate of the initial number. The distribution also provides a credible interval for this estimated initial number. The best estimate and credible interval are shown to be robust against small perturbations of the ranges and therefore can help assessors achieve consensus where hard knowledge is scant. The calculation procedure has been automated and made freely downloadable as a Wolfram Demonstration.

Evaluation of a stochastic inactivation model for heat-activated spores of Bacillus spp.

Heat activates the dormant spores of certain Bacillus spp., which is reflected in the "activation shoulder" in their survival curves. At the same time, heat also inactivates the already active and just activated spores, as well as those still dormant. A stochastic model based on progressively changing probabilities of activation and inactivation can describe this phenomenon. The model is presented in a fully probabilistic discrete form for individual and small groups of spores and as a semicontinuous deterministic model for large spore populations. The same underlying algorithm applies to both isothermal and dynamic heat treatments. Its construction does not require the assumption of the activation and inactivation kinetics or knowledge of their biophysical and biochemical mechanisms. A simplified version of the semicontinuous model was used to simulate survival curves with the activation shoulder that are reminiscent of experimental curves reported in the literature. The model is not intended to replace current models to predict dynamic inactivation but only to offer a conceptual alternative to their interpretation. Nevertheless, by linking the survival curve's shape to probabilities of events at the individual spore level, the model explains, and can be used to simulate, the irregular activation and survival patterns of individual and small groups of spores, which might be involved in food poisoning and spoilage.

Stochastic and deterministic model of microbial heat inactivation.

Microbial inactivation is described by a model based on the changing survival probabilities of individual cells or spores. It is presented in a stochastic and discrete form for small groups, and as a continuous deterministic model for larger populations. If the underlying mortality probability function remains constant throughout the treatment, the model generates first-order ("log-linear") inactivation kinetics. Otherwise, it produces survival patterns that include Weibullian ("power-law") with upward or downward concavity, tailing with a residual survival level, complete elimination, flat "shoulder" with linear or curvilinear continuation, and sigmoid curves. In both forms, the same algorithm or model equation applies to isothermal and dynamic heat treatments alike. Constructing the model does not require assuming a kinetic order or knowledge of the inactivation mechanism. The general features of its underlying mortality probability function can be deduced from the experimental survival curve's shape. Once identified, the function's coefficients, the survival parameters, can be estimated directly from the experimental survival ratios by regression. The model is testable in principle but matching the estimated mortality or inactivation probabilities with those of the actual cells or spores can be a technical challenge. The model is not intended to replace current models to calculate sterility. Its main value, apart from connecting the various inactivation patterns to underlying probabilities at the cellular level, might be in simulating the irregular survival patterns of small groups of cells and spores. In principle, it can also be used for nonthermal methods of microbial inactivation and their combination with heat.

Modeling the efficacy of triplet antimicrobial combinations: yeast suppression by lauric arginate, cinnamic acid, and sodium benzoate or potassium sorbate as a case study.

The growth of four spoilage yeasts, Saccharomyces cerevisiae, Zygosaccharomyces bailii, Brettanomyces bruxellensis, and Brettanomyces naardenensis, was inhibited with three-agent (triplet) combinations of lauric arginate, cinnamic acid, and sodium benzoate or potassium sorbate. The inhibition efficacy was determined by monitoring the optical density of yeast cultures grown in microtiter plates for 7 days. The relationship between the optical density and the sodium benzoate and potassium sorbate concentrations followed a single-term exponential decay model. The critical effective concentration was defined as the concentration at which the optical density was 0.05, which became an efficacy criterion for the mixtures. Critical concentrations of sodium benzoate or potassium sorbate as a function of the lauric arginate and cinnamic acid concentrations were then fitted with an empirical model that mapped three-agent combinations of equal efficacy. The contours of this function are presented in tabulated form and as two- and three-dimensional plots. Triplet combinations were highly effective against all four spoilage yeasts at three practical pH levels, especially at pH 3.0. The triplet combinations were particularly effective for inhibiting growth of Z. bailii, and combinations containing potassium sorbate had synergistic activities. The equal efficacy concentration model also allowed tabulation of the cost of the various combinations of agents and identification of those most economically feasible.

Probabilistic model of microbial cell growth, division, and mortality.

After a short time interval of length deltat during microbial growth, an individual cell can be found to be divided with probability Pd(t)deltat, dead with probability Pm(t)deltat, or alive but undivided with the probability 1-[Pd(t)+Pm(t)]deltat, where t is time, Pd(t) expresses the probability of division for an individual cell per unit of time, and Pm(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat's properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population's size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a "shoulder," and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With Pd(t) and Pm(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.

Isothermal and non-isothermal kinetic models of chemical processes in foods governed by competing mechanisms.

A process or reaction that peaks at high temperatures but not at low ones indicates competition between synthesis and degradation. A proposed phenomenological model composed of a decay factor superimposed on a growth term can describe both. Temperature elevation shortens the two subprocesses' characteristic times and increases their rates. The degradation's characteristic time relative to the experiment's determines whether a peak is observed. All of the parameters determine the peak's height and shape as can be seen in two interactive Wolfram demonstrations on the Web. Detailed knowledge of the underlying mechanisms is unnecessary for the model's construction, and uniqueness is not a prerequisite either. However, different expressions might be needed for ongoing processes and ones initially undetectable. The model's applicability is demonstrated with published results on very different reactions in foods. In principle, it can be converted into a dynamic rate equation for simulating a process's evolution under non-isothermal conditions.

Estimating the heat resistance parameters of bacterial spores from their survival ratios at the end of UHT and other heat treatments.

Accurate determination of bacterial cells or the isothermal survival curves of spores at Ultra High Temperatures (UHT) is hindered by the difficulty in withdrawing samples during the short process and the significant role that the come up and cooling times might play. The problem would be avoided if the survival parameters could be derived directly from the final survival ratios of the non-isothermal treatments but with known temperature profiles. Non-linear inactivation can be described by models that have at least three survival parameters. In the simplified version of the Weibullian -log logistic model they are n, representing the curvature of the isothermal semilogarithmic survival curves, T(c), a marker of the temperature where the inactivation accelerates and k, the slope of the rate parameter at temperatures well above T(c). In principle, these three unknown parameters can be calculated by solving, simultaneously, three rate equations constructed for three different temperature profiles that have produced three corresponding final survival ratios, which are determined experimentally. Since the three equations are constructed from the numerical solutions of three differential equations, this might not always be a practical option. However, the solution would be greatly facilitated if the problem could be reduced to the solution of only two simultaneous equations. This can be done by progressively changing the value of n by small increments or decrements and solving for k and T(c). The iterations continue until the model constructed with the calculated k and T(c) values correctly predicts the survival ratio obtained in a third heat treatment with a known temperature profile. Once n, k, and T(c) are established in this way, the resulting model can be used to predict the complete survival curves of the organism or spore under almost any contemplated or actual UHT treatment in the same medium. The potential of the method is demonstrated with simulated inactivation patterns and its predictive ability with experimental survival data of Bacillus sporothermodurans. Theoretically at least, the shown calculation procedure can be applied to other thermal preservation methods and to the prediction of collateral biochemical reactions, like vitamin degradation or the synthesis of compounds that cause discoloration. The concept itself can also be extended to non-Weibullian inactivation or synthesis patterns, provided that they are controlled by only three or fewer kinetic parameters.

Prediction of an organism's inactivation patterns from three single survival ratios determined at the end of three non-isothermal heat treatments.

Traditionally, an organism's heat resistance parameters have been determined from a set of experimental isothermal survival data. Sometimes, however, even approximating an isothermal profile, and/or obtaining counts at sufficiently short time intervals, is extremely difficult for technical and logistic reasons. The problem would be avoided if the survival parameters could be calculated from the final survival ratios determined at the end of non-isothermal heat treatments with known temperature profiles. Theoretically, if the heat resistance were characterized by three unknown survival parameters, they could be extracted by solving three simultaneous dynamic survival curves' equations. In practice, because of the three equation's complexity - they are themselves the numerical solutions of three differential rate equations - and because the experimental final survival ratios might have a scatter, realistic estimates of the survival parameters require short cut and averaging methods for their calculation. Such a method has been tried with published dynamic inactivation data on Salmonella enteritidis and Escherichia coli. The concept was validated by the ability of the Weibullian-Log logistic model, whose three survival parameters had been obtained directly from final experimental survival ratios only, to predict entire non-isothermal survival curves that had not been used in the model's formulation. The methodology need not be restricted to Weibullian and simpler survival patterns but its practicality might be lost if there are more than three survival parameters. In principle, the same procedure can be extended to biochemical processes that occur during heat preservation, especially at very high temperatures. Estimating inactivation kinetic parameters without isothermal data could also facilitate the quantification of microbial survival under realistic processing conditions and in the actual food rather than in a surrogate medium.