Use of the Weibull model for lactococcal bacteriophage inactivation by high hydrostatic pressure

Four lactococcal bacteriophages (phiLl6-2, phiLl35-6, phiLd66-36 and phiLd67-42) in M17 broth were pressurized at 300 and 350 MPa at room temperature and their survival curves were determined at various time intervals. Tailing (monotonic upward concavity) was observed in all survival curves. The resulting non-linear semi-logarithmic survival curves were described by the Weibull model and goodness of fit of this model was investigated. Regression coefficients (R2), root mean square error (RMSE), residual and correlation plots strongly suggested that Weibull model produced a better fit to the data than the traditional linear model. Hazard plots suggested that the Weibull model was fully appropriate for the data being analyzed. These results have confirmed that the Weibull model, which is mostly utilized to describe the inactivation of bacterial cells or spores by heat and pressure, could be successfully used in describing the lactococcal bacteriophage inactivation by high hydrostatic pressure.     

Estimating the survival of Clostridium botulinum spores during heat treatments

A recently published study of the inactivation of Clostridium botulinum spores at various temperatures in the range of 101 to 121 degrees C and neutral pH revealed that their semilogarithmic survival curves all had considerable upward concavity. This finding indicated that heat inactivation of the spores under these conditions did not follow a first-order kinetics and that meaningful D values could not be calculated. The individual survival curves could be described by the cumulative form of the Weibull distribution, i.e., by log S = -b(T)t(n(T)), where S is the survival ratio and b(T) and n(T) are temperature-dependent coefficients. The fact that at all temperatures in the above range n(T) was smaller than 1 suggested that as time increases sensitive members of the population parish and survivors with increasing resistance remain. If damage accumulation is not a main factor, and the inactivation is path independent, then survival curves under monotonously increasing temperature can be constructed using a relatively simple model, which can be used to calculate the spores' survival in a limiting case. This is demonstrated with computer-simulated heating curves and the experimental constants of the C. botulinum spores, setting the number of decades reduction to 8, 10, and 12 (the current criterion for commercial sterility).     

Survival curves of heated bacterial spores: effect of environmental factors on Weibull parameters

The classical D-value of first order inactivation kinetic is not suitable for quantifying bacterial heat resistance for non-log linear survival curves. One simple model derived from the Weibull cumulative function describes non-log linear kinetics of micro-organisms. The influences of environmental factors on Weibull model parameters, shape parameter "p" and scale parameter "delta", were studied. This paper points out structural correlation between these two parameters. The environmental heating and recovery conditions do not present clear and regular influence on the shape the parameter "p" and could not be described by any model tried. Conversely, the scale parameter "delta" depends on heating temperature and heating and recovery medium pH. The models established to quantify these influences on the classical "D" values could be applied to this parameter "delta". The slight influence of the shape parameter p variation on the goodness of fit of these models can be neglected and the simplified Weibull model with a constant p-value for given microbial population can be applied for canning process calculations.     

Effect of dissolved carbon dioxide on thermal inactivation of microorganisms in milk

Postpasteurization addition of CO2 inhibits growth of certain microorganisms in dairy products, but few workers have investigated the effect of CO2 on the thermal inactivation of microorganisms during pasteurization. Concentrations of CO2 ranging from 44 to 58 mM added to raw whole milk significantly (P < 0.05) reduced the number of surviving standard plate count (SPC) organisms in milk heated over the range of 67 to 93 degrees C. A decrease in thermal survival rates (D-values) for Pseudomonas fluorescens R1-232 and Bacillus cereus ATCC 14579 spores in milk was positively correlated with CO2 concentrations (1 to 36 mM). D(50 degrees C)-values for P. fluorescens significantly decreased (P < 0.05) in a linear fashion from 14.4 to 7.2 min. D(89 degrees C)-values for B. cereus spores were significantly (P < 0.05) decreased from 5.56 min in control milk to 5.29 min in milk containing 33 mM CO2. The Weibull function was used as a model to describe the thermal inactivation of P. fluorescens, B. cereus spores, and SPC organisms in raw milk. Nonlinear parameters for the Weibull function were estimated, and survival data fitted to this model had higher R2 values than when fitted to the linear model, further providing support that the thermal inactivation of bacteria does not always follow first-order reaction rate kinetics. These results suggest that CO2 could be used as a processing aid to enhance microbial inactivation during pasteurization.     

Stochastic and Deterministic Model of Microbial Heat Inactivation

ABSTRACT: 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.     

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 resistances 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 distributions of resistances as a first approximation, although alternative distribution functions can also be used. Changes in growing or environmental condition shift the resistances distribution's 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.     

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.     

Evaluation of selected mathematical models to predict the inactivation of Listeria innocua by pulsed electric fields

The inactivation of Listeria innocua ATCC 51742 by pulsed electric fields was investigated at 35, 40 and 45 kV/cm. Results indicate that at treatment times shorter than 37 μs at 40 and 45 kV/cm, and 49 μs at 35 kV/cm, there is a linear relationship between the logarithm of the survivor fraction and the treatment time. However, longer times result in an abrupt increase in the slope of the inactivation curve and in inactivation values greater than six logarithmic cycles. A model based on Weibull's survival function was used to describe microbial inactivation and then compared to a first-order kinetic model. Distribution parameters of Weibull's survival function and kinetic constant for the first-order kinetic model were calculated by fitting experimental data. Calculated mean times for microbial inactivation from Weibull's distribution were 11.55, 8.65 and 5.39 μs at 35, 40 and 45 kV/cm, respectively. The goodness-of-fit between experimental and predicted values was determined using an accuracy factor. The model based on the Weibull survival distribution provided better accuracy factors than first-order kinetics. The model based on Weibull's survival function seems promising for describing survival curves that exhibit concavity.     

A PREDICTIVE MODEL FOR HIGH-PRESSURE CARBON DIOXIDE INACTIVATION OF MICROORGANISMS

The Weibull model, which is commonly used for thermal inactivation of microorganisms in literature, was used to describe microbial inactivation by high-pressure carbon dioxide (HPCD). The number of parameters of the model was reduced from two to one in order to avoid interrelationship of these parameters with a slight loss of goodness-of-fit. A second-order polynomial function fulfilling a number of constraints was proposed for the secondary modeling of the time-constant parameter of the reduced Weibull model. This function consists of both pressure and temperature and therefore can be used for HPCD treatments.

PRACTICAL APPLICATIONS

The application of any new technology in food preservation requires a reliable model that accurately describes and predicts the inactivation data of microorganisms. In principle, the methodology presented here could be used to describe and predict the survival data for high-pressure carbon dioxide inactivation of microorganisms at least for some pressure and temperature ranges if the isobaric/isothermal survival curves of these microorganisms are linear, concave upward or downward.     

Calculation of the total lethality of conductive heat in cylindrical cans sterilization using linear and non linear survival kinetic models

There is growing evidence suggesting that inactivation of bacterial spores may follow the Weibullian model, of which the log linear relationship between survival ratio and time is just a special case. Where true, the time and not only the temperature dependence of the rate must be taken into account. Also, it is of interest to estimate spores survival not only at the coldest point but also throughout the whole container's volume.Numerical calculation on changing survival ratios was performed by combining the conductive heat transfer model with that of the nonlinear inactivation kinetics. Results are based on published linear and non linear inactivation parameters of Clostridium botulinum spores, typical thermal properties of solid foods and realistic thermal processing conditions. Results could be used to quantify the efficacy of thermal processes of solid products in terms of spores' residual survival ratio at the coldest point and in the container as a whole.     

Combined pressure-thermal inactivation kinetics of Bacillus amyloliquefaciens spores in egg patty mince

Bacillus amyloliquefaciens is a potential surrogate for Clostridium botulinum in validation studies involving bacterial spore inactivation by pressure-assisted thermal processing. Spores of B. amyloliquefaciens Fad 82 were inoculated into egg patty mince (approximately 1.4 x 10(8) spores per g), and the product was treated with combinations of pressure (0.1 to 700 MPa) and heat (95 to 121 degrees C) in a custom-made high-pressure kinetic tester. The values for the inactivation kinetic parameter (D), temperature coefficient (zT), and pressure coefficient (zP) were determined with a linear model. Inactivation parameters from the nonlinear Weibull model also were estimated. An increase in process pressure decreased the D-value at 95, 105, and 110 degrees C; however, at 121 degrees C the contribution of pressure to spore lethality was less pronounced. The zP-value increased from 170 MPa at 95 degrees C to 332 MPa at 121 degrees C, suggesting that B. amyloliquefaciens spores became less sensitive to pressure changes at higher temperatures. Similarly, the zT-value increased from 8.2 degrees C at 0.1 MPa to 26.8 degrees C at 700 MPa, indicating that at elevated pressures, the spores were less sensitive to changes in temperature. The nonlinear Weibull model parameter b increased with increasing pressure or temperature and was inversely related to the D-value. Pressure-assisted thermal processing is a potential alternative to thermal processing for producing shelf-stable egg products.     

Quantifying the effects of heating temperature, and combined effects of heating medium pH and recovery medium pH on the heat resistance of Salmonella typhimurium

The influence of heating treatment temperature, pH of heating and recovery medium on the survival kinetics of Salmonella typhimurium ATCC 13311 is studied and quantified. From each non-log linear survival curve, Weibull model parameters were estimated. An average shape parameter value of 1.67 was found, which is characteristic of downward concavity curves and is in agreement with values estimated from other S. typhimurium strains. Bigelow type models quantifying the heating temperature, heating and recovery medium pH influences are fitted on scale parameter delta data (time of first decimal reduction), which reflects the bacterial heat resistance. The estimate of z(T) (4.64 degrees C) is in the range of values given in the literature for this species. The influence of pH of the heating medium on the scale parameter (z(pH): 8.25) is lower than that of the recovery pH medium influence (z(')(pH): 3.65).     

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.     

Modeling the inactivation of Bacillus subtilis spores during cold plasma sterilization

Cold plasma sterilization is an emerging non-thermal technology that is receiving great attention in the food processing area. Plasma is a neutral ionized gas composed of reactive gas species that inactivate bacteria or spores in a variety of food materials without compromising the main physico-chemical characteristics of the food. Survival curves of Bacillus subtilis spores were obtained after spore strip samples containing an initial spore population of 1.5–2.5 × 10⁶ cfu/strip were subjected to plasma treatment. The shape of the survival curves was clearly not linear indicating that spores exhibited a spectrum of inactivation resistances to the plasma treatment. A Weibull model was used to describe these curves. In order to capture the effects of the typical variability in the concentration of the inactivating reactive gas species during plasma processing, time-varying concentrations were incorporated in the calculating approach. The result was an ordinary differential equation (ODE) that was numerically solved using MATLAB. This approach was successfully applied to describe the survival of B. subtilis spores during plasma processing as well as data obtained from the literature for B. atrophaeus. Ozone was assumed the lethal reactive gas species responsible for spore inactivation.Industrial relevanceModeling plasma processing is of great interest because it may provide an accurate estimation of time and conditions required for a complete plasma-based sterilization process.     

A new kinetic model for thermal inactivation of microorganisms: development and validation using Escherichia coli O157:H7 as a test organism.

A new kinetic model has been proposed to simulate the nonlinear behavior of survivor curves frequently observed in thermal inactivation of microorganisms. This model incorporates a time component into the first-order inactivation kinetics and is capable of describing the linear, convex, and concave survivor curves. The model was validated using Escherichia coli O157:H7 as a test microorganism. Ground beef (93% lean) samples inoculated to 10(7) to 10(8) CFU/g of meat were subjected to immersion heating at 55, 57.5, 60, 62.5, and 65 degrees C, respectively, in a water bath. All the survivor curves in this study showed upward concavity. Linear and nonlinear regressions were used to fit the survivor curves to the linear first-order inactivation kinetics and the proposed model. Analyses showed that the new kinetic model provides a much better estimate of the thermal inactivation behavior of E. coli O157:H7 in ground beef.     

Modeling the inactivation of Salmonella Typhimurium, Listeria monocytogenes, and Salmonella Enteritidis on poultry products exposed to pulsed UV light

Pulsed UV light inactivation of Salmonella Typhimurium on unpackaged and vacuum-packaged chicken breast, Listeria monocytogenes on unpackaged and vacuum-packaged chicken frankfurters, and Salmonella Enteritidis on shell eggs was explained by log-linear and Weibull models using inactivation data from previous studies. This study demonstrated that the survival curves of Salmonella Typhimurium and L. monocytogenes were nonlinear exhibiting concavity. The Weibull model was more successful than the log-linear model in estimating the inactivations for all poultry products evaluated, except for Salmonella Enteritidis on shell eggs, for which the survival curve was sigmoidal rather than concave, and the use of the Weibull model resulted in slightly better fit than the log-linear model. The analyses for the goodness of fit and performance of the Weibull model produced root mean square errors of 0.059 to 0.824, percent root mean square errors of 3.105 to 21.182, determination coefficients of 0.747 to 0.989, slopes of 0.842 to 1.042, bias factor values of 0.505 to 1.309, and accuracy factor values of 1.263 to 6.874. Overall, this study suggests that the survival curves of pathogens on poultry products exposed to pulsed UV light are nonlinear and that the Weibull model may generally be a useful tool to describe the inactivation patterns for pathogenic microorganisms affiliated with poultry products.     

Comparison of the thermal inactivation of Bacillus subtilis spores in foods using the modified Weibull and Bigelow equations

The association of a modified Weibull model and Bigelow model was applied to the thermal inactivation of Bacillus subtilis spores heated in phosphate buffer, milk, kayu (a Japanese style rice porridge) and soy sauce as well. The inactivation kinetics presented a light downward concave profile, the acidic pH increased the efficiency of the heat treatment but on the opposite, lesser the water activity, weaker was the efficiency. The heat treatment kinetics observed in milk, soy sauce and kayu were greatly different from each other, while no large difference between sterilized whole milk, UHT whole milk, sterilized skim milk and UHT skim milk, were observed. The model established in buffer system allowed heat treatment in milk products to be simulated although it could not be employed to describe the inactivation of B. subtilis spores in soy sauce and kayu. For these two latter products, the food itself had to be introduced in the model as a parameter. Finally, this approach combining primary model (to simulate inactivation kinetics) and secondary model (to introduce temperature, pH, a_w and food matrix effect) seemed available for food application, nevertheless validations of results such as challenge-tests, must be performed before it is put to routine use.     

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 inactivation kinetics of food borne pathogens at a constant temperature

Four food borne pathogens (Listeria monocytogenes CA and Ohio₂, Salmonella enteritidis FDA and Salmonella typhimurium E21274, Escherichia coli O157:H7 931 and 933, Staphylococcus aureus 485 and 765) were inactivated under mild temperature (60 °C) and their survival curves determined at selected time intervals. Tailing was observed in all survival curves as a monotonic upward concavity. The resulting survival curves were either described by the Weibull or traditional first-order model and goodness of fit of these models was investigated. Regression coefficients (R²), root mean square error (RMSE) and correlation plots suggested that Weibull model produced a better fit to the data than the traditional model. Hazard plots suggested that the Weibull model was fully appropriate for the data being analysed. Although more studies should be carried out to evaluate the applicability of the nonlinear models, the present study has shown that thermal process calculations should most probably be reconsidered. This could lead to a reduction in under- and over-processing of thermally treated foods