Mark D. Normand

Generating microbial survival curves during thermal processing in real time

Aims:  To develop a method to calculate and record theoretical microbial survival curves during thermal processing of foods and pharmaceutical products simultaneously with the changing temperature. Moreover, to demonstrate that the method can be used to calculate nonisothermal survival curves, with widely available software such as Microsoft Excel®.

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 models 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(Treference)] = c(T-Treference). 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 models equation rather than to the existence of a temperature independent “energy of activation.” If T+273.16°C in the Arrhenius models equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and Treference, 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(Treference) = a[1/ (Treference+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.

Comparison of the Acoustic and Mechanical Signatures of Two Cellular Crunchy Cereal Foods at Various Water Activity Levels

Cheese balls and croutons at various water activity levels (0·11–0·75) were compressed between parallel metal plates using a Universal testing machine and their acoustic emissions were recorded at compact disc quality (sampling rate of 44·1 kHz). The sound wave record (up to about 2 MB) had high intensity bursts at irregular intervals. These records were compressed after the background noise had been filtered out to produce files of less than 48 kB. The compressed signatures were characterised by their mean and peak amplitude, two measures of the sound emission intensity, and by the amplitudes standard deviation and the mean magnitude of the power spectrum, two measures of the acoustic signature complexity. All four parameters could be used to monitor the plasticisation effect of water. Although their magnitudes were correlated, they did not always change in unison upon moisture sorption. The standard deviation and mean magnitude of the power spectrum of the compressed acoustic signatures were only broadly correlated with their correspondents in the normalised mechanical signatures primarily because they were not determined from the same particles and because the latter, for technical reasons, were sampled at the low rate of 6 Hz.

Estimating microbial inactivation parameters from survival curves obtained under varying conditions—The linear case

When the isothermal semi-logarithmic survival curves of heat inactivated microbial cells or spores are known to be linear it is possible to calculate their survival parameters from curves obtained under nonisothermal conditions, provided that the temperature history (’profile’) satisfies certain simple mathematical requirements. These requirements have been identified. The concept was tested by retrieving the survival parameters of a Listeria-like organism from generated survival curves for linear and nonlinear heating profiles on which noise had been superimposed. The availability of such a procedure eliminates the need to determine the survival parameters under perfect isothermal conditions, which are difficult to create for technical reasons. It will also enable determination of the survival parameters in the actual medium of interest, which may contain particles or may be too viscous to be treated in a capillary or narrow tube as is currently done. The method can also be used to assess survival parameters in nonthermal inactivation. A treatment with a dissipating chemical agent or anti-microbial is an example. In principle, the concept can be extended to the more general situation where the isothermal or iso-concentration semi-logarithmic survival curves are clearly nonlinear, but this will require a modification of the model and a different numerical calculation procedure.

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, Tc, a marker of the temperature where the inactivation accelerates and k, the slope of the rate parameter at temperatures well above Tc. 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 Tc. The iterations continue until the model constructed with the calculated k and Tc values correctly predicts the survival ratio obtained in a third heat treatment with a known temperature profile. Once n, k, and Tc 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.

Description of normal, log—normal and Rosin—Rammler particle populations by a modified version of the beta distribution function

Abstract Data generated by the normal, log—normal and Rosin—Rammler distribution functions were normalized and fitted with a slightly modified version of the beta distribution function. As long as the frequency function had a zero or practically zero value at the two ends of a finite size range, the fitted curves were, for all practical purposes, indistinguishable from the normal and Rosin—Rammler distributions. The fit of the modified beta function to narrow log—normal distributions was also excellent but it declined significantly as the distribution spread increased. It appears, though, that for real particle populations, having a finite size range, and not necessarily a perfectly smooth size distribution, the modified beta function can replace all three functions, thus providing a way to present and compare the different size distribution patterns in terms of a single mathematical expression.

Probabilistic Model of Microbial Cell Growth, Division, and Mortality

ABSTRACT After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability Pd(t)δt, dead with probability Pm(t)δt, or alive but undivided with the probability 1 − [Pd(t) + Pm(t)]δt, 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 habitats 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 populations 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.

Estimating the frequency of high microbial counts in commercial food products using various distribution functions

Industrial microbial count records usually form an irregular fluctuating time series. If the series is truly random or weakly autocorrelated, the fluctuations can be considered as the outcome of the interplay of numerous factors that promote or inhibit growth. These factors usually balance each other, although not perfectly, hence, the random fluctuations. If conditions are unchanged, then at least in principle the probability that they will produce a coherent effect, i.e., an unusually high (or low) count of a given magnitude, can be calculated from the count distribution. This theory was tested with miscellaneous industrial records (e.g., standard plate count, coliforms, yeasts) of various food products, including a dairy-based snack, frozen foods, and raw milk, using the normal, log normal, Laplace, log Laplace, Weibull, extreme value, beta, and log beta distribution functions. Comparing predicted frequencies of counts exceeding selected levels with those actually observed in fresh data assessed their efficacy. No single distribution was found to be inherently or consistently superior. It is, therefore, suggested that, when the probability of an excessive count is estimated, several distribution functions be used simultaneously and a conservative value be used as the measure of the risk.

Determination of elasticity of gels by successive compression-decompression cycles

Abstract Cylindrical specimens of agar, alginate and kappa carrageenan gels (1 and 2.5%) were subjected to five successive compression-decompression cycles to pre-determined strains. The total and irrecoverable work in these cycles, and their dependence on the strain level and concentration, were characteristic of the gel type. The portion of the irrecoverable work, which served as a measure of deviation from ideal elasticity, could not be correlated with the strength of the gels or their deformability (failure strain) or stiffness, indicating that the ‘degree of elasticity’ is an independent property of these gels.

A distribution function for particle populations having a finite size range and a mode independent of the spread

Abstract A distribution function and its cumulative form are described and their mathematical properties analyzed. Since f(0) = f(a) = 0, this function can be applied to real particle populations, having a definite size range 0 a 2 , μ > a 2 or μ a 2 , respectively. This permits the simulation of extreme changes in particle size distribution patterns without changing the model format. The parameter B is an arbitrary constant whose effect on the distribution curve shape is of little significance as long as its magnitude remains small relative to that of a. Although the integrals that appear in f(x) and F(x) cannot be easily solved analytically, their values can conveniently be computed by standard numerical methods employing a microcomputer.