Modelling thermal inactivation of Listeria monocytogenes in sucrose solutions of various water activities

The heat resistance of Listeria monocytogenes was determined in sucrose solutions with water activity (a(w)) ranging from 0.99 to 0.90. At all temperatures investigated shape of the survival curves depended on the a(w) of the treatment medium. The survival curves for a(w)=0.99 appeared to be linear, for a(w)=0.96 were slightly upwardly concaved and for a(w)=0.93 and 0.90 were markedly concave upward. A mathematical model based on the Weibull distribution provided a good fit for all the survival curves obtained in this investigation. The effect of the temperature and a(w) on the Weibull model parameters was also studied. The shape parameter (p) depended on the a(w) of the treatment medium but in each medium of different a(w) the temperature did not have a significant effect on this parameter. The p parameter followed a linear relationship with a(w). The scale parameter (delta) decreased with the temperature following an exponential relationship and increased by decreasing the a(w) in the range from 0.99 to 0.93. However the delta parameter of survival curves obtained at a(w)=0.90 were lower than those obtained at a(w)=0.93. A mathematical model based on the Weibull parameters was built to describe the joint effect of temperature and a(w) on thermal inactivation of L. monocytogenes. This model provides a more complete information on the influence of the a(w) on the L. monocytogenes than the data initially generated. The model developed indicated that the effect of the a(w) on the thermal resistance of L. monocytogenes varied depending upon the temperature of treatment.     

Predicting heat inactivation of Listeria monocytogenes under nonisothermal treatments

The aim of this study was to find a model that accurately predicts the heat inactivation of Listeria monocytogenes (ATCC 15313) at constantly rising heating rates (0.5 to 9 degrees C/min) in media of different pH values (4.0 to 7.4). Survival curves of L. monocytogenes obtained under isothermal treatments at any temperature were nearly linear. Estimations of survival curves under nonisothermal treatments obtained from heat resistance parameters of isothermal treatments adequately fit experimental values obtained at pH 4.0. On the contrary, survivors were much higher than estimations at pH 5.5 and 7.4. The slower the heating rate and the longer the treatment time, the greater the differences between the experimental and estimated values. An equation based on the Weibullian-like distribution, log S(t) = (t/delta)p, accurately described survival curves of L. monocytogenes obtained under nonisothermal conditions within the range of heating rates investigated. A nonlinear relationship was observed between the scale parameter (delta) and the heating rate, which allowed the development of an equation capable of predicting the inactivation rate of L. monocytogenes under nonisothermal treatments at pH 5.5 and 7.4. The model predictions were a good fit to the measured data independent of the magnitude of the thermotolerance increase. This work might contribute to the increase in safety of those food products that require long heating lag phases during the pasteurization process.     

Calculating microbial survival parameters and predicting survival curves from non-isothermal inactivation data

Irrespective of whether the isothermal semi-logarithmic survival curves of heat inactivated microbial cells or spores are linear or nonlinear, it is theoretically possible to numerically calculate their survival parameters from inactivation data obtained under non-isothermal conditions. A method to do the calculation, when the temperature history ('profile') is expressed algebraically, is demonstrated with simulated survival curves. It has been tested with the published survival data of Salmonella, whose nonlinear semi-logarithmic isothermal survival curves can be described by a power law model. The reported survival ratios of Salmonella, determined during non-isothermal heat treatments in a broth and in ground chicken breast, were used to estimate its isothermal survival parameters in the two media and their temperature dependence. These, in turn, were used to predict the cells' survival curves under different temperature 'profiles.' There was a good agreement between the predicted and the reported experimental survival curves in the broth case and reasonable agreement in the ground chicken breasts, where the database was considerably smaller The development of a mathematical method to calculate survival parameters from non-isothermal inactivation data will eliminate the need to determine these parameters under isothermal conditions, which can only be approximated and are technically difficult to perform. In many cases, the proposed method will also enable the determination of the survival parameters in the actual food or medium of interest, which may contain particles, or is too viscous to be heated and cooled effectively using the currently available experimental procedures. In principle, the described mathematical method can also be used to assess organisms' survival parameters in nonthermal inactivation processes, such as exposure to a dissipating chemical agent or the application of ultra high-pressure.     

Effect of a previous heat shock on the thermal resistance of Listeria monocytogenes and Pseudomonas aeruginosa at different pHs

In this work we study the effect of heat shocks of various durations up to 60 min, at different temperatures between 35 and 45 degrees C, in media of pH 4.0, 5.5 and 7.4 on the heat resistance of Listeria monocytogenes and Pseudomonas aeruginosa. The pattern of survival curves after heat treatment did not change with the application of a previous heat shock. However, the kinetics of inactivation was different for the two microorganisms studied. Whereas the inactivation of L. monocytogenes was similar to an exponential function of heating time and therefore straight survival curves were obtained, survival curves corresponding to P. aeruginosa showed convex profiles. All survival curves obtained in this investigation were fitted to Weibull-based Mafart equation: log(10)S(t)=-(t / delta)(p). The magnitude of the heat shock induced thermotolerance increased with treatment medium pH. At pH 7.4 the increase in heat tolerance depended on the duration and temperature of the heat shock. On the contrary, at pH 5.5 and pH 4.0, the heat-shock temperature did not exert any effect. The observed maximum delta values increased 2.3, 4.0 and 9.3 fold for L. monocytogenes, and 1.3, 2.1 and 8.4 fold for P. aeruginosa, at pH 4.0, 5.5 and 7.4, respectively. This research has proven that Mafart equation allows studying and quantifying the effect of heat shocks on bacterial heat resistance.     

On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model

A simple and parsimonious model which originated from the Weibull frequency distribution was proposed to describe nonlinear survival curves of spores. This model was suitable for downward concavity curves (Bacillus cereus and Bacillus pumilus), as well as for upward concavity curves (Clostridium botulinum). It was shown that traditional F values calculated from this new model were no longer additive, to such an extent that a heat treatment should be better characterized by the obtained decimal reduction of spores. A modified Bigelow method was then proposed to assess this decade reduction or to optimize the heat treatment for a target reduction ratio.     

COMBINED EFFECT OF HEAT AND ALKALI IN STERILIZING SUGARCANE BAGASSE

ABSTRACT— Bacterial spores contained in sugarcane bagasse were subjected to various combinations of heat exposure and alkali concentration and the rate of destruction determined for each set of conditions. A series of survival, thermal death time and alkaline destruction curves revealed a different mode of death by heat exposure than alkali treatment. Addition of alkali into the heating menstruum caused the death rates of bacterial spores to be much greater than with heat alone at a given temperature. Exposure of the spores to a temperature of 75°C for 130 min was required to reduce the spore population by 90% with heat treatment alone. Incorporation of a 1% NaOH solution into the heating menstruum effected the same degree of destruction of the spores within a 2 min period at the same temperature. From a series of thermal destruction and alkaline destruction curves, an empirical equation expressing the relationship between the death rate of bacterial spores, and the intensity of temperature and the concentration of alkali was established. The equation reveals that the death rate of bacterial spores is affected in an exponential manner by temperature and in a direct relationship by alkali concentration. Using the equation, sterilization time for various combinations of temperature and alkali concentration was determined and the overall correlation index between the experimental data and computed value was 0.877.     

A quantitative approach for studying the effect of heat treatment conditions on resistance and recovery of Bacillus cereus spores.

When a bacterial population undergoes an unfavourable transient increase in temperature, a death phase followed by a lag and growth phase are observed for the surviving and cultivable population. The lag phase is of great interest in regard to food safety, but for bacterial spores, very few studies have been carried out on the evolution of lag time versus heat treatment duration. The experiments monitored on spores of two strains of Bacillus cereus showed a biphasic evolution of the lag time for germination of stressed spores with increase in heat treatment duration, at 90 degrees C, 95 degrees C and 100 degrees C and for different recovery conditions in laboratory medium. Initially, the lag time increased with the duration of the thermal stress up to an optimum. Subsequently, the lag time decreased, with increasing stress duration, to a threshold. The evolution of the lag time to germination of surviving spores, under the range of temperature tested, was similar to the evolution of the lag time to growth after mild-temperature treatments observed with vegetative cells of Escherichia coli and Listeria monocytogenes by Breand et al. (1997) [Breand, S., Fardel, G., Flandrois, J.-P., Rosso, L., Tomassone, R., 1997. A model describing the relationship between lag time and mild temperature increase duration. Int. J. Food Microbiol. 38, 157-167]. The mathematical model established from these investigations could also be applied in the case of heat-stressed spores. Moreover, it was observed that there was a significant linear relationship between the D-value and the treatment duration that gave the maximum lag time.     

INACTIVATION KINETICS AND REDUCTION OF BACILLUS COAGULANS SPORE BY THE COMBINATION OF HIGH PRESSURE AND MODERATE HEAT

The combination effect of high pressure (400, 500 and 600 MPa) and moderate heat (70 and 80C) on the inactivation kinetics and reduction of Bacillus coagulans spore in phosphate buffer and ultra-high temperature (UHT) whole milk was investigated. The pressure come-up time and corresponding logarithmic reduction of spore inactivation were considered during pressure-thermal treatment. B. coagulans spore had a much higher resistance to pressure in UHT whole milk than in phosphate buffer. Survival data were modeled using the linear, Weibull and log-logistic models to obtain relevant kinetic parameters. The tailing phenomenon occurred in all survival curves, indicating the linear model was not adequate for describing these curves. The mean square error and regression coefficient suggested that the log-logistic model produced best fits to all survival curves, followed by the Weibull model.

PRACTICAL APPLICATIONS

It becomes increasingly apparent that high-pressure treatment combined with moderate heat treatment for low acid and acid products is often required for effective bacterial spores' inactivation. Consequently, the prediction model of microbial survival curves is essential. Bacillus coagulans is a slightly pressure-resistant and relatively heat-resistant spoilage bacterium of considerable concern during the processing of acid foods. Spore inactivation effect during the pressure come-up time is sometimes considerable and should not beignored. The use of mathematical models to predict inactivation for spores could help the food industry further to develop optimum process conditions.     

Quantifying the combined effects of the heating time, the temperature and the recovery medium pH on the regrowth lag time of Bacillus cereus spores after a heat treatment

The purpose of this study was to quantify the lag time of re-growth of heated spores of Bacillus cereus as a function of the conditions of the heat treatment: temperature, duration and pH of the recovery medium. For a given heating temperature, curves plotting lag times versus time of heating show more or less complex patterns. However, under a heating time corresponding to a decrease of 2 decimal logarithms of the surviving populations of spores, a linear relationship between the lag time of growth and the time of the previous heat treatment can be observed. The slope of this linear relationship followed itself a Bigelow type linear relationship, the slope of which yielded a zeta-value very close to the observed conventional z-value. It was then concluded that the slope of the regrowth lag time versus the heating time followed a linear relationship with the sterilisation value reached in the course of the previous heat treatment. A sharp effect of the pH of the medium which could be described by a simple "secondary" model was observed. As expected, the observed intercept of the linear relationship between lag time and heating time (lag without previous heating) was dependent on only the pH of the medium and not on the heating temperature.     

A model describing the relationship between regrowth lag time and mild temperature increase for Listeria monocytogenes

In order to comply with the consumer demand for ready-to-eat and look 'fresh' products, mild heat treatment will be used more and more in the agrofood industry. Nonetheless there is no tool to define the most appropriate mild heat treatment. In order to build this tool, it is necessary to study and describe the response of a bacterial population to a mild increase in temperature, from the dynamic point of view. The response to a mild increase in temperature, defined by stress duration and temperature, consisted in a mortality phase followed by the lag time of the survivors and their exponential growth. The effect of the mild increase in temperature on the mortality phase was described in a previous paper (Bréand et al., Int. J. Food Microbiol., in press). The effect of the stress duration on the lag was presented in a previous paper (Bréand et al., Int. J. Food Microbiol. 38 (1997) 157-167). In particular, the biphasic relationship between the lag and the stress duration was observed and modelled with a four parameter nonlinear model: the primary model (Bréand et al., Int. J. Food Microbiol. 38 (1997) 157-167). The study presented in this paper deals with the effect of the stress temperature on the biphasic relationship between the lag time and the stress duration. The secondary models describing the effect of the stress temperature on this biphasic relationship, were empirically built from our experimental data concerning Listeria monocytogenes. This work pointed out that the higher the stress temperature, the narrower the range of stress duration for which the lag time increased. Since the primary and the secondary models of the lag time were available, the global model describing the effect of the mild increase duration and temperature directly on the lag was fitted. This model allowed an improvement of the parameter estimator precision. The potential contribution in mild heat treatment optimization of this global model and the one built for the mortality phase (Bréand et al., Int. J. Food Microbiol., in press) is discussed.     

Thermal inactivation of Campylobacter jejuni in broth

New Zealand has a high rate of reported campylobacteriosis compared with other developed countries. One possible reason is that local strains have greater heat tolerance and thus are better able to survive undercooking; this hypothesis is supported by the remarkably high D-values reported for Campylobacter jejuni in The Netherlands. The objective of this study was to investigate the thermal inactivation of isolates from New Zealand in broth, using strains that are commonly found in human cases and food samples in New Zealand. Typed Campylobacter strains were heated to a predetermined temperature using a submerged-coil heating apparatus. The first-order kinetic model has been used extensively in the calculation of the thermal inactivation parameters, D and z; however, nonlinear survival curves have been reported, and a number of models have been proposed to describe the patterns observed. Therefore, this study compared the conventional first-order model with eight nonlinear models for survival curves. Kinetic parameters were estimated using both one- and two-step regression techniques. In general, nonlinear models fit the individual inactivation data sets better than the log-linear model. However, the log-linear and the (nonlinear) Weibull models were the only models that could be successfully fitted to all data sets. For seven relevant New Zealand C. jejuni strains, at temperatures from 51.5 to 60°C, D- and z-values were obtained, ranging from 1.5 to 228 s and 4 to 5.2°C, respectively. These values are in broad agreement with published international data and do not indicate that the studied New Zealand C. jejuni strains are more heat resistant than other strains, in contrast with some reports from The Netherlands.     

An optimization algorithm for estimation of microbial survival parameters during thermal processing

Isothermal microbial survival curves are usually described by either linear or nonlinear time-dependent models, from which non-isothermal survival curves can be generated if the parameters describing the survival kinetics of the microbial population are known. In order to estimate these parameters, an algorithm based on the steepest decent optimization method was developed. The algorithm searches the values of the survival parameters which minimize the sum of the squared differences between the experimental data and the calculated values provided by the model. The difference of the proposed algorithm with a typical optimization technique is that each data point used is not necessarily coming from the same thermal treatment; instead, data from different non-isothermal processes can be used. The developed algorithm was tested by using published non-isothermal survival data of Salmonella. The data showed that the survival curves can be described by the Weibull model, an already accepted and frequently used nonlinear model. Salmonella's survival parameters were estimated from the end points and all data points, respectively, of three non-isothermal survival curves. The results obtained showed that the number of survival data points must be sufficiently large to obtain true or statistically sound values of the survival parameters. A suitable way to achieve this is to implement the algorithm using all data points of multiple non-isothermal survival curves or a large number of end points of non-isothermal treatments. Mathematically, the developed algorithm should be applicable to any microbial survival kinetics accurately describing the inactivation of the microorganisms because no specific survival kinetics has to be pre-assumed to run the algorithm.     

Interactive software for estimating the efficacy of non-isothermal heat preservation processes

The most commonly used methods to generate microbial inactivation curves are based on the assumptions that microbial mortality follows first order kinetics and that the temperature effect on the 'D value' or exponential rate constant is determined by the log linear model or the Arrhenius equation, respectively. However, many bacterial cells and spores follow the Weibull-Log Logistic (WeLL) model and software to simulate pasteurization and sterilization processes using this model has been available for some years as free downloadable programs written in MS Excel. According to this model, an organism's heat resistance parameters are T(c), a marker of the temperature level where the inactivation accelerates, k, the steepness of the Weibullian rate parameter in the lethal regime where T>T(c) and n, a measure of the semi logarithmic isothermal survival curve's concavity and its direction. Because the traditional first order kinetics is just a special case of the Weibullian model with n=1.0, the software is applicable to both linear and non-linear inactivation. Recently, Wolfram Research Inc., the maker of Mathematica, has made its interactive program Mathematica Player free downloadable software. A user, who need not have a copy of Mathematica, can view and download any of the numerous graphic demonstrations from the Wolfram Demonstrations Project web site, and continuously manipulate their dynamic parameters with sliders on the screen. One set of five such demonstrations allows the user to generate and adjust the temperature profile of heat processes, modify the targeted organism's Weibullian survival parameters and immediately observe the corresponding semi-logarithmic survival curve and the equivalent time at a reference temperature, which can also be manipulated by a slider. This free program enables food microbiologists, technologists and engineers to examine a large number of heat processing options and assess their potential safety implications. It can also serve as a training and educational tool in industry and academia.     

Modeling the combined effect of high hydrostatic pressure and mild heat on the inactivation kinetics of Listeria monocytogenes Scott A in whole milk

The survival curves of Listeria monocytogenes Scott A inactivated by high hydrostatic pressure were obtained at four temperatures (22, 40, 45 and 50 °C) and two pressure levels (400 and 500 MPa) in UHT whole milk. Elevated temperatures substantially promoted the pressure inactivation of L. monocytogenes. A 5-min treatment of 500 MPa at 50 °C resulted in a more than 8-log₁₀ reduction of L. monocytogenes, while at 22 °C a 35-min treatment was needed to obtain the same level of inactivation. Tailing was observed in all survival curves, indicating that the linear model was not adequate for describing these curves. The log-logistic model consistently produced best fits to all survival curves and the modified Gompertz model the poorest. The Weibull model produced fits as good as the log-logistic model at the temperature range of 40–50 °C. The Weibull model provided reasonable predictions of inactivation of L. monocytogenes at temperature levels other than the experimental temperatures; however, the log-logistic model was found to be inferior at predicting inactivation.     

Influence of heat transfer with tube methods on measured thermal inactivation parameters for Escherichia coli

The purpose of this study was to investigate the influence of heat transfer on measured thermal inactivation kinetic parameters of bacteria in solid foods when using tube methods. The bacterial strain selected for this study, Escherichia coli K-12, had demonstrated typical first-order inactivation characteristics under isothermal test conditions. Three tubes of different sizes (3, 13, and 20 mm outer diameter) were used in the heat treatments at 57, 60, and 63 degrees C with mashed potato as the test food. A computer model was developed to evaluate the effect of transit heat transfer behavior on microbial inactivation in the test tubes. The results confirmed that the survival curves of E. coli K-12 obtained in 3-mm capillary tubes were log linear at the three tested temperatures. The survival curves observed under nonisothermal conditions in larger tubes were no longer log linear. Slow heat transfer alone could only partially account for the large departures from log-linear behavior. Tests with the same bacterial strain after 5 min of preconditioning at a sublethal temperature of 45 degrees C revealed significantly enhanced heat resistance. Confirmative tests revealed that the increased heat resistance of the test bacterium in the center of the large tubes during the warming-up periods resulted in significantly larger D-values than those obtained with capillary tube methods.     

柑橘皮在热处理过程中内部温度的时空分布特性

目的：获得柑橘热处理过程中果实内部不同点随时间变化的温度分布规律。方法：从传热的角度研究柑橘皮在热处理过程中的温度变化；提出一种新的方法来计算柑橘皮中的温度传递规律，建立一个新的数学模型来模拟柑橘皮在热处理过程中的温度分布；并重点研究3种柑橘（奉节脐橙、长寿血橙和资阳蜜桔）内部温度场的时空分布。结果：热处理时间随水温的升高呈指数级下降，得到了不同品种柑橘的详细指数关系表达式，并通过数据回归方法提出了柑橘热处理时间与水浴温度之间的函数公式。结论：热处理时间随着水浴温度升高呈指数衰减，过低的水域温度会因传热温差过小导致热处理时间极大延长，增加能源消耗。     

热胁迫和培养温度对大肠杆菌抗热性的影响

研究大肠杆菌O157:H7 ATCC 43889经50、60 ℃和70 ℃反复多次热胁迫处理与在10、28、36 ℃和45 ℃的条件下生长培养后对其80 ℃抗热性的影响。分别对ATCC 43889进行50、60 ℃和70 ℃的热胁迫,研究在一定的热力致死温度条件下杀死某细菌数量90%所需要的时间（D值）的变化,观察ATCC 43889热胁迫前后菌落形态和个体形态的变化;将ATCC 43889置于10、28、36 ℃和45 ℃培养至稳定期,分别测定其在80 ℃的存活量,再利用Weibull模型拟合其在80 ℃的热致死曲线。结果表明,50、60 ℃和70 ℃热胁迫处理均可诱导ATCC 43889抗热性增加,经10 次热胁迫并传代培养后,其D值分别为第1次热胁迫处理后的1.88、2.38 倍和8.18 倍,D值随热处理次数的增加不断增大,说明胁迫温度越高,D值越大,其抗热性越强;经过60 ℃和70 ℃热胁迫后,ATCC 43889菌落形态和个体形态与对照组相比差异显著;在80 ℃,ATCC 43889的致死曲线表明,胁迫温度越高,其抗热性越强（P＜0.05）;在10～45 ℃培养,随培养温度的升高,ATCC 43889的抗热性显著增加（P＜0.05）。利用Weibull模型可以较好地拟合ATCC 43889经过50、60 ℃和70 ℃热胁迫处理10 次后与10、28、36 ℃和45 ℃培养后在80 ℃的抗热性曲线,随着胁迫温度和培养温度的升高,ATCC 43889的抗热性都呈增加趋势。综上,一定热处理与培养温度可胁迫诱导大肠杆菌ATCC 43889抗性热增强和形态的变化。     

Modelling the influence of the sporulation temperature upon the bacterial spore heat resistance, application to heating process calculation

Environmental conditions of sporulation influence bacterial heat resistance. For different Bacillus species a linear Bigelow type relationship between the logarithm of D values determined at constant heating temperature and the temperature of sporulation was observed. The absence of interaction between sporulation and heating temperatures allows the combination of this new relationship with the classical Bigelow model. The parameters zT and zT(spo) of this global model were fitted to different sets of data regarding different Bacillus species: B. cereus, B. subtilis, B. licheniformis, B. coagulans and B. stearothermophilus. The origin of raw products or food process conditions before a heat treatment can lead to warm temperature conditions of sporulation and to a dramatic increase of the heat resistance of the generated spores. In this case, provided that the temperature of sporulation can be assessed, this model can be easily implemented to rectify F values on account of possible increase of thermal resistance of spores and to ensure the sterilisation efficacy.     

Combined use of ultrasound and natural antimicrobials to inactivate Listeria monocytogenes in orange juice

The presence of Listeria monocytogenes could seriously affect the safety of nonpasteurized fruit juices. High-intensity ultrasound combined with mild heat treatment and natural antimicrobials may be an alternative technology for fruit juice preservation. The response of L. monocytogenes in orange juice to combined treatments involving moderate temperature (45 degrees C), high-intensity ultrasound (600 W, 20 kHz, 95.2 microM wave amplitude), and the addition of different levels of vanillin (0, 1,000, 1,500, and 2,000 ppm), citral (0, 75, and 100 ppm), or both was investigated to find the most effective inactivation treatment. Nonlinear semilogarithmic survival curves were successfully fitted by the Weibull model. The presence of vanillin or citral greatly increased the bactericidal effect of thermosonication and changed the distribution of inactivation times. When both antimicrobials were added together and ultrasound was applied, narrowest frequency shapes, skewed to the right, with low ariances and death time means between 1.6 and 2.6 min, were obtained. Orange juices with 1,500 or 1,000 ppm of vanillin and 100 ppm of citral were pleasant for the consumers.     

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 organism's or spore's 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.