Micha Peleg

Reinterpretation of Microbial Survival Curves

The heat inactivation of microbial spores and the mortality of vegetative cells exposed to heat or a hostile environment have been traditionally assumed to be governed by first-order reaction kinetics. The concept of thermal death time and the standard methods of calculating the safety of commercial heat preservation processes are also based on this assumption. On closer scrutiny, however, at least some of the semilogarithmic survival curves, which have been considered linear are in fact slightly curved. This curvature can have a significant effect on the thermal death time, which is determined by extrapolation. The latter can be considerably smaller or larger depending on whether the semilogarithmic survival curve has downward or an upward concavity and how the experimenter chooses to calculate decimal reduction time. There are also numerous reports of organisms whose semilogarithmic survival curves are clearly and characteristically nonlinear, and it is unlikely that these observations are all due to a mixed population or experimental artifacts, as the traditional explanation implies. An alternative explanation is that the survival curve is the cumulative form of a temporal distribution of lethal events. According to this concept each individual organism, or spore, dies, or is inactivated, at a specific time. Because there is a spectrum of heat resistance in the population--some organism or spores are destroyed sooner, or later, than others--the shape of the survival curve is determined by its distributions properties. Thus, semilogarithmic survival curves whether linear or with an upward or a downward concavity are only reflections of heat resistance distributions having a different, mode variance, and skewness, and not of mortality kinetics of different orders. The concept is demonstrated with published data on the lethal effect of heat on pathogens and spores alone and in combination with other factors such as pH or high pressure. Their different survival patterns are all described in terms of different Weibull distribution of resistances as a first approximation, although alternative distribution functions can also be used. Changes in growing or environmental condition shift the resistances distributions mode and can also affect its spread and skewness. The presented concept does not take into account the specific mechanisms that are the cause of mortality or inactivation--it only describes their manifestation in a given microbial population. However, it is consistent with the notion that the actual destruction of a critical system or target is a probabilistic process that is due, at least in part, to the natural variability that exists in microbial populations.

Modeling microbial survival during exposure to a lethal agent with varying intensity.

Traditionally, the efficacy of preservation and disinfection processes has been assessed on the basis of the assumption that microbial mortality follows a first-order kinetic. However, as departures from this assumed kinetics are quite common, various other models, based on higher-order kinetics or population balance, have also been proposed. The database for either type of models is a set of survival curves of the targeted organism or spores determined under constant conditions, that is, constant temperature, chemical agent concentration, etc. Hence, to calculate the outcome of an actual industrial process, where conditions are changing, as in heating and cooling during a thermal treatment or when the agent dissipates as in chlorination or hydrogen peroxide application, one has to integrate the momentary effects of the lethal agent. This involves mathematical models based on assumed mortality kinetics, and simulated or measured history, for example, temperature-time or concentration-time relationships at the “coldest” point. It is shown that the survival curve under conditions where the agent intensity increases, decreases, or oscillates can be constructed without assuming any mortality kinetics and without the use of the traditional D and Z values, which require linear approximation, and without thermal death times, which require extrapolation. The actual survival curves can be compiled from the isothermal survival curves provided that growth and damage repair do not occur over the pertinent time scale and that the mortality rate is a function of only the momentary agent intensity and of the organisms or spores survival fraction (but not of the rate at which this fraction has been reached). The calculation is greatly facilitated if both the “isothermal” survival curves and the time-dependent agent intensity can be expressed algebraically. The differential equation derived from these considerations can be solved numerically to produce the required survival curve under the changing conditions. The concept is demonstrated with simulated survival curves during heating at different rates, heating and cooling cycles, oscillating temperature, and exposure to a dissipating chemical agent. The simulated thermal processes are based on published data of Clostridium botulinum spores, whose semilogarithmic survival curves have upward concavity and on a hypothetical “Listeria-like” organism whose semilogarithmic curves have downward concavity.

Linearization of Relaxation and Creep Curves of Solid Biological Materials

Stress relaxation curves of many solids can be normalized and presented in the linear form of F(0)t/[F(0)−F(t)]=k1+k2t, where F(t) is the decaying force and k1,k2 are constants. The reciprocal of k2 denotes the asymptotic value of the relaxed portion of the initial stress. Since in active biological materials equilibrium conditions in the conventional sense are difficult to determine, the asymptotic values can be used to calculate residual moduli that are representative of the material short‐term rheological characteristics. Similarly, creep curves of solids in which the strain e(t) stabilizes or practically stabilizes with time (e.g., under small loads or in compression) can be presented by t/e(t)=k1+k2t, where 1/k2 is the asymptotic strain. It is shown that the relationships between asymptotic moduli, so calculated, and the strain in relaxation or the stress in creep can carry information that is relevant to structural changes that occur during deformation of the material such as development of hydrosta...

Comparison of two methods for stress relaxation data presentation of solid foods

Published exponential relaxation equations, derived from Maxwellian models, were used to generate data for linear representation in the form ofP(0) ·t/(P(0) —P(t)) =k1 +k2t whereP(t) is the decaying parameter (force, stress or modulus),P(0) its initial value (att = 0) andk1 andk2 constants. The computer plots indicated that the fit of this normalized and linearized form was excellent for equations containing at least three exponential decay terms. The fit was not as good for some of the two-term exponential equations mainly due to the lack of accurate account for the initial stage of the relaxation process. In all the cases, however, the linear representation could clearly reveal the general rheological character of the analysed materials in terms of the relative degree of solidity.

Microbial survival curves - The reality of flat shoulders and absolute thermal death times

Abstract Occasionally, experimental survival curves of micro-organisms exposed to a lethal agent have a flat region and traditionally it has been interpreted as evidence of the existence of a “shoulder”. However, if the survival curve is considered the cumulative distribution of lethal events, which reflects a spectrum of resistances, or sensitivities, then when the distributions mean, or mode, is large relative to its spread, a region resembling a “shoulder” will be observed irrespective of whether the distribution is symmetric or skewed. Computer simulated survival curves generated with the Fermi and Weibull distributions as models demonstrate that the shape of the survival curve alone is, therefore, insufficient to confirm any specific inactivation mechanisms at the cellular and molecular level, although it can refute the existence of some. Microbial mortality has also been assumed to be a process following an exponential decay and hence that a certain degree of survival is inevitable. It is not inconceivable, however, that there can be an absolute thermal death time if the survivors are being progressively weakened by a prolonged exposure to the lethal agent. This testable possibility is demonstrated with simulated survival curves generated with two mathematical models.

On calculating sterility in thermal and non-thermal preservation methods

Abstract There is growing evidence that the mortality of microbial cells, and the inactivation of bacterial spores, exposed to a hostile environment need not follow a first order kinetics. Consequently microbial semi-logarithmic survival curves are frequently non-linear, and their shape can change with temperature or under different chemical agent concentrations, for example. Experimental semi-logarithmic survival curves under unchanging conditions, can be described by an equation whose coefficients are determined by the particular temperature, agent concentration, etc. If the dependency of these coefficients on temperature, agent concentration, etc., can be expressed algebraically, then in principle one can construct the survival curve for the changing or transient conditions that exist in industrial thermal and non-thermal treatments. This is done by incorporating the lethal agents mode of change, e.g. the heating or pressure curve into the survival curve equation parameters. The result is a mathematical model that would enable the calculation of the time needed to achieve any degree of microbial survival ratio numerically, without the need to assume any mortality kinetics. Such a model can be used to assess, or compare, the efficacy of different preservation processes where the intensity of the lethal agent changes with time. The concept is demonstrated with a special simple case using simulated thermal treatments. The outcome of the simulations is presented as planar log survival vs time relationships and as curves in a three-dimensional log survival–temperature–time or log survival–concentration–time space.

Calculating Salmonella Inactivation in Nonisothermal Heat Treatments from Isothermal Nonlinear Survival Curves

Salmonella cells in two sugar-rich media were heat treated at various constant temperatures in the range of 55 to 80 degrees C and their survival ratios determined at various time intervals. The resulting nonlinear semilogarithmic survival curves are described by the model log10S(t) = -b(T)tn(T), where S(t) is the momentary survival ratio N(t)/N0, and b(T) and n(T) are coefficients whose temperature dependence is described by two empirical mathematical models. When the temperature profile, T(t), of a nonisothermal heat treatment can also be expressed algebraically, b(T) and n(T) can be transformed into a function of time, i.e., b[T(t)] and n[T(t)]. If the momentary inactivation rate primarily depends on the momentary temperature and survival ratio, then the survival curve under nonisothermal conditions can be constructed by solving a differential equation, previously suggested by Peleg and Penchina, whose coefficients are expressions that contain the corresponding b[T(t)] and n[T(t)] terms. The applicability of the model and its underlying assumptions was tested with a series of eight experiments in which the Salmonella cells, in the same media, were heated at various rates to selected temperatures in the range of 65 to 80 degres C and then cooled. In all the experiments, there was an agreement between the predicted and observed survival curves. This suggests that, at least in the case of Salmonella in the tested media, survival during nonisothermal inactivation can be estimated without assuming any mortality kinetics.

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.