When is simple good enough: a comparison of the Gompertz,Baranyi, and three-phase linear models for fitting bacterial growth curves

The use of primary mathematical models with curve fitting software is dramatically changing quantitative food microbiology. The two most widely used primary growth models are the Baranyi and Gompertz models. A three-phase linear model was developed to determine how well growth curves could be described using a simpler model. The model divides bacterial growth curves into three phases: the lag and stationary phases where the specific growth rate is zero (gm=0), and the exponential phase where the logarithm of the bacterial population increases linearly with time (gm=constant). The model has four parameters: N_o(Log₈of initial population density), NMAX(Log₈of final population density), tLAG(time when lag phase ends), and tMAX(time when exponential phase ends). A comparison of the linear model was made against the Baranyi and Gompertz models, using established growth data forEscherichia coli0157:H7. The growth curves predicted by the three models showed good agreement. The linear model was more ‘robust' than the others, especially when experimental data were minimal. The physiological assumptions underlying the linear model are discussed, with particular emphasis on assuring that the model is consistent with bacterial behavior both as individual cells and as populations. It is proposed that the transitional behavior of bacteria at the end of the lag phase can be explained on the basis of biological variability.     

Comparison of the Baranyi model with the modified Gompertz equation for modelling thermal inactivation of Listeria monocytogenes Scott A

The Baranyi model was used to fit the four commonly observed survival curves: linear curves, those with a lag phase, those with a tailing phase and sigmoidal curves. It was validated by using published experimental data for thermal inactivation of Listeria monocytogenes Scott A heated in infant formula and compared with the modified Gompertz equation. For the prediction performance, the Baranyi model was better and more robust than the modified Gompertz equation.     

Growth kinetics of Listeria monocytogenes in broth and beef frankfurters--determination of lag phase duration and exponential growth rate under isothermal conditions

ABSTRACT:  The objective of this study was to develop a new kinetic model to describe the isothermal growth of microorganisms. The new model was tested with Listeria monocytogenes in tryptic soy broth and frankfurters, and compared with 2 commonly used models—Baranyi and modified Gompertz models. Bias factor (BF), accuracy factor (AF), and root mean square errors (RMSE) were used to evaluate the 3 models. Either in broth or in frankfurter samples, there were no significant differences in BF (approximately 1.0) and AF (1.02 to 1.04) among the 3 models. In broth, the mean RMSE of the new model was very close to that of the Baranyi model, but significantly lower than that of the modified Gompertz model. However, in frankfurters, there were no significant differences in the mean RMSE values among the 3 models. These results suggest that these models are equally capable of describing isothermal bacterial growth curves. Almost identical to the Baranyi model in the exponential and stationary phases, the new model has a more identifiable lag phase and also suggests that the bacteria population would increase exponentially until the population approaches to within 1 to 2 logs from the stationary phase. In general, there is no significant difference in the means of the lag phase duration and specific growth rate between the new and Baranyi models, but both are significantly lower than those determined from the modified Gompertz models. The model developed in this study is directly derived from the isothermal growth characteristics and is more accurate in describing the kinetics of bacterial growth in foods.     

生孢梭菌孢子生长与失活模型的统一化研究

为探讨预测微生物生长和失活的预测模型统一化问题,研究将生孢梭菌孢子热失活“镜像化”曲线用描述微生物生长的Gompertz 模型和Baranyi 模型进行模拟,并通过标准预测误对两种模型进行比较。实验表明：两种模型都能较好的模拟生孢梭菌孢子热失活,但t 检验分析差异不显著,标准预测误比较表明Gompertz 模型优于Baranyi 模型。建议用Gompertz 模型统一描述生孢梭菌孢子生长和失活情况。     

Modeling the inactivation of Salmonella typhimurium by dense phase carbon dioxide in carrot juice

The inactivation of Salmonella typhimurium inoculated into acidified carrot juice subjected to dense phase carbon dioxide (DPCD) was investigated. The pressures in the study were 10, 20 and 30 MPa, the temperatures were 32, 37 and 42 °C, and the treatment time was 5–90 min. The inactivation effect of DPCD was enhanced by increasing pressure and temperature. The sigmoid inactivation curves were characterized with the lag phase, exponential inactivation phase, and resistant phase. The inactivation curves were fitted to the modified Gompertz equation and the modified Logistic equation, the modified Gompertz equation was superior since its lowest residual sum of squares (RSS) was lower although there was no significant difference of goodness-of-fit between both models as indicated by F-test. The λ (the duration of the lag phase) and t_4-D (the time necessary to achieve 4-log cycles reduction) decreased with increasing pressure or temperature. The k_dm (the maximum specific value of the inactivation rate, min⁻¹) increased with increasing temperatures, and decreased with increasing pressures. The activation energy (Ea) and the activation volume (Va) necessary for inactivating S. typhimurium by DPCD were 19.06–29.39 kJ mol⁻¹ and 18.89–58.27 cm³ mol⁻¹.     

Synergistic effect of carbon dioxide atmospheres and high hydrostatic pressure to reduce spoilage bacteria on poultry sausages

Synergistic effect of combining high hydrostatic pressure (HHP) and CO₂ atmospheres has been studied against Leuconostoc carnosum, Brochothrix thermosphacta, Salmonella enteritidis, Campylobacter jejuni and Listeria innocua separately inoculated in poultry sausages. The microbial counts of the HHP treated samples (350 MPa 10 min at room temperature) and CO₂ atmosphere packaged were compared with non pressure treated and air packaged ones, analyzed at 20 h and 7 days after the treatment. The results showed a synergistic effect of the combination of these two preserving technologies against all the microorganisms studied, except in S. enteritidis which showed a greater resistance under CO₂ atmospheres, and C. jejuni, that is especially sensitive at high pressures. It seems that cell damage produced by high pressure facilitates the penetration of carbon dioxide into the microorganisms' cells, affecting their metabolism and consequently their growth. Using CO₂ atmospheres in combination with HHP treatments, pressure could be lowered without compromising the reduction of microbial counts.     

Influence of Pressurization Rate and Mode on Inactivation of Natural Microorganisms in Purple Sweet Potato Nectar by High Hydrostatic Pressure

High hydrostatic pressure (HHP) processing was applied for inactivation of natural microorganisms, including total aerobic bacteria (TAB), and yeasts and molds (Y&M) in purple sweet potato nectar (PSPN). The pressurization rates were 60 and 120 MPa/min, the pressurization modes were stepwise and linear, the pressure-holding times were 2.5–25 min, and the pressure levels were 400–600 MPa. In all the experimental conditions, the Y&M in PSPN were not detected, the maximum reduction of TAB was greater than 4 log cycles, and the inactivation of TAB was closely related to the HHP parameters. The fast pressurization rate and linear pressurization mode enhanced the inactivation effect of HHP on TAB. With increasing the pressure level and pressure-holding times, the inactivation of TAB was also enhanced. The mathematical models were fitted to the inactivation kinetic data of TAB and fitness of these models was investigated; the Weibull and Biphasic model successfully fitted all the inactivation curves. Pressurization rate and mode had a significant impact on the parameters of the models.     

A Gompertz Model Approach to Microbial Inactivation Kinetics by High‐Pressure Processing (HPP): Model Selection and Experimental Validation

A recently proposed Gompertz model (GMPZ) approach describing microbial inactivation kinetics by high‐pressure processing (HPP) incorporated the initial microbial load (N₀) and lower microbial quantification limit (N_lim), and simplified the dynamic effects of come‐up time (CUT). The inactivation of Listeria innocua in milk by HPP treatments at 300, 400, 500, and 600 MPa and pressure holding times (t_hold) ≤10 min was determined experimentally to validate this model approach. Models based on exponential, logistic‐exponential, and inverse functions were evaluated to describe the effect of pressure on the lag time (λ) and maximum inactivation rate (μ_max), whereas the asymptote difference (A) was fixed as A = log₁₀(N₀/N_lim). Model performance was statistically evaluated and further validated with additional data obtained at 450 and 550 MPa. All GMPZ models adequately fitted L. innocua data according to the coefficient of determination (R² ≥ 0.95) but those including a logistic‐exponential function for μ_max(P) were superior (R² ≥ 0.97). These GMPZ versions predicted that approximately 597 MPa is the theoretical pressure level (P_λ) at which microbial inactivation begins during CUT, mathematically defined as λ (P = P_λ) = t_CUT, and matching the value observed on the microbial survival curve at 600 MPa. As pressure increased, predictions tended to slightly underestimate the HPP lethality in the tail section of the survival curve. This may be overseen in practice since the observed microbial counts were below the predicted log₁₀ N values. Overall, the modeling approach is promising, justifying further validation work for other microorganisms and food systems.     

Mathematical modeling on the growth of Staphylococcus aureus in sandwich

The growth of Staphylococcus aureus in sandwich fillings at different incubation temperatures was tested. These growth data were fitted into the Gompertz model, Logistic model, and Baranyi model in order to compare the goodness-of-fit of the 3 primary models using several factors such as coefficient of determination (R²), the standard deviation (S_y.x), and the Akaike’s information criterion (AIC). The Gompertz model showed the best statistical fit. Hence, growth parameters such as specific growth rate (SGR) and lag time (LT) obtained from the Gompertz model were used to construct the secondary models. Further, developed models were evaluated by bias factor (B_f) and accuracy factor (A_f). For the SGR, the Bf value was 0.993 and A_f value was 1.156 which indicated conservative predictions. While for LT, a clear deviation was observed between predictions and observations (B_f=0.635 and A_f=1.592). The results, however, were also considered acceptable after comparing with previous publications.     

Estimating the maximum growth rate from microbial growth curves: definition is everything

The maximum growth rate (μ_max) is an important parameter in modelling microbial growth under batch conditions. However, there are two definitions of this growth parameter in current use and some of the comparisons of data made in the literature fail to acknowledge this important fact.We compared values of μ_max obtained by applying the Gompertz, logistic and Baranyi–Roberts models to experimental data on the growth of Listeria monocytogenes and Listeria innocua using both absorbance and viable counts measurements of cell concentration. All three models fitted the experimental data well, however, the values of μ_max obtained using the Gompertz and logistic models were similar to each other but substantially different from those predicted by the Baranyi–Roberts model. The latter growth model was used to derive a second estimate of μ_max based on the slope at the inflection point of the growth curve function; this value was in closer agreement with those obtained using the Gompertz or logistic models. Conditions were identified when values of μ_max based on different definitions would converge towards one another.     

Predictive Modeling of the Growth of Lactobacillus Viridescens under Non-isothermal Conditions

Food spoilage by microorganisms is a major problem that can generate large economic losses to industries, making critical the application of technologies for predicting shelf life, aiming to obtain products with higher quality. The Lactic Acid Bacteria (LAB), including Lactobacillus viridescens, are among the main groups of microorganisms responsible for spoilage of refrigerated meat products, vacuum packed and under modified atmosphere. The growth of the LAB can be predicted by mathematical models, which describe the influence of various environmental factors (such as non-isothermal conditions) on microbial growth. The objective of this study was to obtain a mathematical model able to predict the growth of L. viridescens in non-isothermal conditions in culture medium (MRS broth). Six isothermal growth curves (at 4, 8, 12, 16, 20 and 30̊C) were described by Baranyi and Roberts model and the dependence of maximum specific growth rate (μ_max) parameter on the temperature was described by square root secondary model. The model was validated using L. viridescens experimental data in the temperature ranging from 6 to 10°C and 5 to 11°C, changing every 12 and 24h, respectively. The results showed that it was possible to predict safely (bias factor greater than 1) the growth of L. viridescens in MRS broth under non-isothermal conditions. The observed prediction deviations may have been caused by abrupt temperature changes, generating intermediate adaptation phases.     

Modelling the effect of environmental factors on the growth of Aspergillus parasiticus and mycotoxin production in paddy during storage

The simulated experiment of A. parasiticus isolated from the paddy was carried out during the paddy storage for 20 days. The growth and mycotoxin data were collected for constructing kinetic and probability models of moulds. The Baranyi and Gompertz model was employed as the primary model and estimated the lag phase and maximum specific growth rate. Secondary models, such as polynomial, Davey and Gibson model were used and completely evaluated under different conditions. The polynomial equation was highly rated compared with Gibson and Davey model and gave realistic temperatures and a_w for mould growth. Logistic model showed promising results on the prediction of growth boundary and AFB₁ production. Employed models showed promising predicted results, indicating that it is an effective tool for describing and predicting the growth of moulds under different temperatures and a_w. The results can be applied to develop the optimal strategy to prevent fungal spoilage and aflatoxin production during paddy storage.     

A Gompertz Model Approach to Microbial Inactivation Kinetics by High-Pressure Processing Incorporating the Initial Counts,Microbial Quantification Limit,and Come-Up Time Effects

During come-up time (CUT), the time to reach a desired processing pressure, isobaric-isothermal conditions cannot be assumed in the estimation of kinetic parameters for the design of commercial high-pressure processing (HPP) treatments. Since CUT effects on microbial population, enzyme activity, and chemical concentration are often ignored, kinetic models incorporating the non-isobaric and non-isothermal conditions prevailing during CUT were the objective of this work. The analysis of peer-reviewed data on the HPP inactivation of bacteria (counts observations n = 919, 60 survival curves) and bacterial spores (n = 273, 12 curves) showed that a Gompertz model (GMPZ) approach is an effective alternative. The GMPZ parameter A was fixed as the difference between the initial population (log₁₀ N _o ) and the lower quantification limit of microbial counts (log₁₀ N _lim), while exponential equations were used to describe pressure effects on the lag time (λ) and the maximum inactivation rate (μ_max). In low-acid media (pH > 4.5), λ decreased exponentially with pressure, allowing the identification of a theoretical pressure level (P _λ) sufficient to initiate microbial inactivation during CUT. The parameter μ_max exponentially increased with pressure for all evaluated datasets. Dynamic pressure effects during CUT were simplified by assuming isobaric conditions during CUT (t _CUT), allowing to obtain GMPZ parameter estimates using only nonlinear regression (R ² ～ 0.938, σ ² = 0.030–0.604). The proposed approach is a simpler, promising tool for a more informative analysis of the kinetics of microbial inactivation by HPP and should be further validated with additional experimental data.     

Inactivation of Bacillus cereus spores using a combined treatment of UV-TiO2 photocatalysis and high hydrostatic pressure

Bacillus cereus spores are resistant to high hydrostatic pressure (HHP) processing treatment. A combination of UV-TiO₂ photocatalysis (UVTP for 10 min) and two cycles of 600 MPa HHP treatment for 10 min for the first cycle and 1 min for the second cycle (UVTP-2HHP) at ambient temperature was applied to inactivate B. cereus spores inoculated on a solidified agar matrix (SAM) used as a model matrix. Two cycles of HHP treatment were used as a strategy for induction of spore germination, followed by inactivation. UVTP and 2 cycles of HHP resulted in a 5.0-log CFU/cm² spore reduction (initial spore count was 6.6 log CFU/cm²), including an approximate 0.8-log CFU/cm² reduction due to a synergistic effect. The inactivation mechanism of UVTP pretreatment was related to lipid peroxidation of the spore membrane based on the level of malondialdehyde (MDA) making spores susceptible to the HHP treatment. Flow cytometry and transmission electron microscopic (TEM) analyses showed severe physiological alteration and structural damage to spores after the combined treatment. UVTP and 2 cycles of HHP showed potential for effective inactivation of B. cereus to ensure food safety from B. cereus spores on food products.Practical applicationsInactivation of bacterial spores remains a technical challenge for HHP and other interventions because spores are highly resistant to high pressure. Pretreatment with UVTP followed by two cycles of HHP resulted in reduction in B. cereus spores due to a synergistic effect. This hurdle technology of UVTP and HHP can help food industry in ensuring food safety against the Bacillus spores.     

Modeling the influence of water activity and ascospore age on the growth of Neosartorya fischeri in pineapple juice

The ascospores of resistant fungi, Neosartorya fischeri, can survive commercial pasteurization, diminishing the shelf life of these products. The time that the ascospores remain in the environment and the effect that they can cause on mold growth are still unknown. This study is aimed to evaluate the influence of water activity (a_w) from 0.90 to 0.99 and the ascospore age (I) from 30 to 90 days of vitro incubation on the growth of N. fischeri in pineapple juice by mathematical modeling. The growth parameters on pineapple juice: adaptation phase (λ), maximum specific growth rate (μ_max) and maximum diameter reached by the colony (A) were obtained by fitting Modified Gompertz and Logistic models to the experimental data. Both models were able to describe microbial growth in pineapple juice, but the Modified Gompertz model presented a slightly superior performance based on statistical indices (correlation coefficients (R²), mean square error (MSE), Bias Factor and Accuracy Factor). The minimum values of λ and A, calculated by the Modified Gompertz model, were 64.7 h and 6.3 mm, while the maximum values were 178.2 h and 20.8 mm, respectively. The result showed that ascospore age did not influence the growth but a_w was statistically significant to the growth parameters λ and A.     

Mathematical models to predict kinetic behavior and aflatoxin production of Aspergillus flavus under various temperature and water activity conditions

Mathematical models were developed to predict fungal growth and aflatoxin production of Aspergillus flavus. Fungal growth and aflatoxin concentrations were measured. The Baranyi model was fitted to fungal growth and toxin production data to calculate kinetic parameters. Quadratic polynomial and Gaussian models were then fitted to μ_max and LPD (lag phase duration) values. The ranges of temperature and a _w values showing a μ_max value increase were 15–35°C and 0.891–0.984, respectively. LPD was only observed when the temperature was 20–35°C with a _w=0.891?0.972. The μ_{max growth} value increased up to 35°C with \(b_w = 0.2\left( {b_w = \sqrt {1 - a_w } } \right)\) , then values declined. LPD_growth values increased as the b _w value increased. The μ_max value for aflatoxins increased up to 25°C, but decreased after 30°C, indicating that the developed models are useful for describing the kinetic behavior of Aspergillus flavus growth and aflatoxin production.     

Statistical evaluation of mathematical models for microbial growth

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO(2) (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.     

<Emphasis Type="Italic">Listeria</Emphasis> Inactivation by the Combination of High Hydrostatic Pressure and Lactocin AL705 on Cured-Cooked Pork Loin Slices

The effect of the bacteriocin lactocin AL705 in combination with high hydrostatic pressure (HHP) on the inactivation of Listeria innocua 7, a nonpathogenic indicator for Listeria monocytogenes, deliberately inoculated (ca. 6.4 log CFU/g) onto the surface of ready-to-eat (RTE) sliced cured-cooked pork loin, was evaluated. Nontreated pork slices (control) and treatments subjected to lactocin AL705 (105 AU/ml) and/or HHP (400 or 600 MPa) were prepared. L. innocua 7 was monitored at days 1, 20, and 40 of storage at 4 °C. The results showed a complete inhibition of L. innocua 7 after the combined treatment with lactocin AL705 and 600 MPa and no regrowing of cells up to 40-day storage. The treatment at 600 MPa alone was not enough to avoid regrowth of L. innocua. Ultrastructural cell damage was observed at the cytoplasm and cell membrane/wall levels with all treatments; however, complete cell lysis was observed only with the combined treatment. HHP in combination with lactocin AL705 provided a wider margin of safety as post-processing listericidal treatment of RTE cured-cooked meat products.     

Growth kinetics of Listeria monocytogenes and spoilage microorganisms in fresh-cut cantaloupe

The main objective of this study was to investigate the growth kinetics of Listeria monocytogenes and background microorganisms in fresh-cut cantaloupe. Fresh-cut cantaloupe samples, inoculated with three main serotypes (1/2a, 1/2b, and 4b) of L. monocytogenes, were incubated at different temperatures, ranging from 4 to 43 °C, to develop kinetic growth models. During storage studies, the population of both background microorganisms and L. monocytogenes began to increase almost immediately, with little or no lag phase for most growth curves. All growth curves, except for two growth curves of L. monocytogenes 1/2a at 4 °C, developed to full curves (containing exponential and stationary phases), and can be described by a 3-parameter logistic model. There was no significant difference (P = 0.28) in the growth behaviors and the specific growth rates of three different serotypes of L. monocytogenes inoculated to fresh-cut cantaloupe. The effect of temperature on the growth of L. monocytogenes and spoilage microorganisms was evaluated using three secondary models. For L. monocytogenes, the minimum and maximum growth temperatures were estimated by both the Ratkowsky square-root and Cardinal parameter models, and the optimum temperature and the optimum specific growth rate by the Cardinal parameter model. An Arrhenius-type model provided more accurate estimation of the specific growth rate of L. monocytogenes at temperatures <4 °C. The kinetic models developed in this study can be used by regulatory agencies and food processors for conducting risk assessment of L. monocytogenes in fresh-cut cantaloupe, and for estimating the shelf-life of fresh-cut products.     

Effect of Low-pressure Cold Plasma (LPCP) on the Wettability and the Inactivation of <Emphasis Type="Italic">Escherichia coli</Emphasis> and <Emphasis Type="Italic">Listeria innocua</Emphasis> on Fresh-Cut Apple (<Emphasis Type="Italic">Granny Smith</Emphasis>) Skin

The aim of this study was to investigate the effect of low-pressure cold plasma (LPCP) on the inactivation of Escherichia coli and Listeria innocua on fresh-cut apple skin and its influence on wettability. Cold plasma treatments have shown to be effective to decontaminate foods, but their effect on the wettability has not been well studied. Surface-inoculated apple samples were treated with argon (Ar), nitrogen (N2), oxygen (O₂), and argon-oxygen (Ar-O₂) cold plasma using a commercial LPCP unit. Three different models were used to fit bacterial survival curves after the LPCP treatments. Changes in surface wettability were also determined by measuring the contact angle. The LPCP treatments using Ar, O₂, or Ar-O₂ mixture for 20 min were the most effective to inactivate E. coli with O₂, while the LPCP treatment with N2 for 20 min reduced L. innocua the most for (p?<?0.05). The highest increase in surface wettability was observed in samples treated for 20 min with O₂ and Ar-O₂. Different LPCP treatments have not great effectivity on the inactivation of E. coli and L. innocua on fresh-cut apple surface, but the all treatments changed the surface wettability of apples, making it more hydrophilic. This can be considered as a negative effect of the LPCP treatment because it can facilitate the adhesion and proliferation of re-contaminating microorganisms. More research should be undertaken to explore the use of other gases and complex surfaces.