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Some examples of, and some problems with, the use of nonlinear logistic regression in predictive food microbiology
Abstract
A new technique, nonlinear logistic regression, is described for modelling binomially distributed data, i.e., presence/absence data where growth is either observed or not observed, for applications in predictive food microbiology. Some examples of the successful use of this technique are presented, where the controlling factors are temperature, water activity, pH and the concentration of lactic acid, a weakly dissociating organic acid. Generally speaking, good-fitting models were obtained, as evidenced using various performance measures and goodness-of-fit statistics. As may be expected with a new statistical technique, some problems were encountered with the implementation of the modelling approach and these are discussed.
... Essentially the same approach was adopted by Lanciotti et al. (2001) to develop G/NG models for B. cereus, S. aureus, and Salmonella enteritidis. Ratkowsky (2002) commented on the increased exibility in being able to determine all of the parameters in the model during the regression, and subsequent studies developed the approach, eventually leading to a novel nonlinear logistic regression technique Tienungoon et al., 2000). Ratkowsky (2002) pointed out that nonlinear logistic regression was a new statistical technique and discussed bene ts and problems with that approach specifically in relation to growth limits modeling. ...
... Ratkowsky (2002) commented on the increased exibility in being able to determine all of the parameters in the model during the regression, and subsequent studies developed the approach, eventually leading to a novel nonlinear logistic regression technique Tienungoon et al., 2000). Ratkowsky (2002) pointed out that nonlinear logistic regression was a new statistical technique and discussed bene ts and problems with that approach specifically in relation to growth limits modeling. A problem with models of the form of Equation 3.44 is that for conditions more extreme than the parameters corresponding to T min , pH min , a w min , etc., and which are tested experimentally though not expected to permit growth, the terms containing those parameters would become negative. ...
... This, in turn, affects the values of the parameters of the fitted model. Ratkowsky (2002) comments that an objective method for selection and deletion of such data is necessary, but does not yet exist. Bolton and Frank (1999) extended the binary logistic regression approach by recoding growth and no growth data to allow a third category: survival, or stasis. ...
... Essentially the same approach was adopted by Lanciotti et al. (2001) to develop G/NG models for B. cereus, S. aureus, and Salmonella enteritidis. Ratkowsky (2002) commented on the increased exibility in being able to determine all of the parameters in the model during the regression, and subsequent studies developed the approach, eventually leading to a novel nonlinear logistic regression technique Tienungoon et al., 2000). Ratkowsky (2002) pointed out that nonlinear logistic regression was a new statistical technique and discussed bene ts and problems with that approach specifically in relation to growth limits modeling. ...
... Ratkowsky (2002) commented on the increased exibility in being able to determine all of the parameters in the model during the regression, and subsequent studies developed the approach, eventually leading to a novel nonlinear logistic regression technique Tienungoon et al., 2000). Ratkowsky (2002) pointed out that nonlinear logistic regression was a new statistical technique and discussed bene ts and problems with that approach specifically in relation to growth limits modeling. A problem with models of the form of Equation 3.44 is that for conditions more extreme than the parameters corresponding to T min , pH min , a w min , etc., and which are tested experimentally though not expected to permit growth, the terms containing those parameters would become negative. ...
... This, in turn, affects the values of the parameters of the fitted model. Ratkowsky (2002) comments that an objective method for selection and deletion of such data is necessary, but does not yet exist. Bolton and Frank (1999) extended the binary logistic regression approach by recoding growth and no growth data to allow a third category: survival, or stasis. ...
... In this case, based on available secondary models presented in Sect. 3.3, one finds that the square root model could be applied to fit the observed growth for the suboptimal temperature range (Ratkowsky 1983). ...
... ffiffiffiffiffiffiffi ffi 1=l p ) assumption as reflected by Ratkowsky (2004). Concerning the maximum growth rate, many works have successfully used the root-square transformation (i.e., ffiffiffi m p ) and log-transformation (e.g., ln (m)) for this parameter (Ratkowsky 2004(Ratkowsky , 1983. ...
... Its simplicity and easy interpretation make it suitable for a first approach to the fitted model. Also, RMSE is a valid index for both linear and nonlinear mathematical functions (Ratkowsky 1983(Ratkowsky , 2004. A low RMSE value indicates a good fitting to data as a result of the closeness of the data points to the fitted model. ...
One of the most critical steps when generating a predictive model is to correctly design an experiment and collect suitable microbial data. Experimental design will influence model structure and validation conditions. The survival and growth of microorganisms in foods is affected not only by the chemical composition of the food and its storage conditions but also by the food matrix. In this sense, a better quantification of the food structure effect has been studied throughout these years. Regarding the method of data collection, although plating count has been widely used (and still is used), there are rapid methods to obtain reliable and cost-effective data. These achievements were primarily based on turbidimetry, although other methods (microscopy, image analysis, flow cytometry, etc.) have arisen as novel approaches in the predictive microbiology field. These aspects are further discussed in this chapter.
... Unfortunately, these nonlinear-appearing parameters make the estimation process of the parameters difficult. Besides, stable solutions are not systematically obtained and convergence to an optimum is often not achieved (Ratkowsky, 2002). Fixing these values at levels derived from square-root-type kinetic models can facilitate the regression process , which is, however, not readily available in statistic software packages. ...
... Conditions beyond and at the theoretical growth limits can not be used for parameter estimation because natural logarithms of negative or zero values are than encountered. Therefore, data can be discarded (Ratkowsky, 2002) or repositioned at X = X min + ε where X represents the environmental factor considered, X min the respective theoretical growth limit and ε a small number to reposition the data before the theoretical growth limit (Gysemans et al., 2007b). In general, no difference will be found because conditions which exceed the theoretical growth limit show 'no growth' and are repositioned near the theoretical growth limit (also 'no growth' region). ...
... The latter model type has the advantage to contain biological meaningful parameters. These parameters, however, appear in a non-linear way hampering parameter estimation (Ratkowsky, 2002). The detailed data gathered in this study, show that buffer systems with acetic and lactic acid, ...
... According to Ratkowsky (2002), the most promising approaches to develop probability models are to use the logit of P, where logit ( P)=ln( P/(1ÀP)) and P is the probability of growth. Ratkowsky and Ross (1995) proposed a model to relate the probability of growth of Shigella flexneri as a function of temperature, pH, water activity and sodium nitrite: ...
... More recently, Ratkowsky (2002) proposed the inclusion of square and cross products terms in the model. In this work, we also investigated a more general form of Eq. ...
... This latest method was used by Ratkowsky and Ross (1995) and Presser et al. (1998) to model the probability of growth of S. flexneri and E. coli, respectively. Ratkowsky (2002) discussed the problems encountered with the use of non-linear regression. Although estimating rather than fixing the non-linear parameters improves the performance of the model, stable solutions are not systematically obtained and convergence to an optimum is often not achieved (Ratkowsky, 2002). ...
Data from a database on microbial responses to the food environment (ComBase, see www.combase.cc) were used to study the boundary of growth several pathogens (Aeromonas hydrophila, Escherichia coli, Listeria monocytogenes, Yersinia enterocolitica). Two methods were used to evaluate the growth/no growth interface. The first one is an application of the Minimum Convex Polyhedron (MCP) introduced by Baranyi et al. [Baranyi, J., Ross, T., McMeekin, T., Roberts, T.A., 1996. The effect of parameterisation on the performance of empirical models used in Predictive Microbiology. Food Microbiol. 13, 83-91.]. The second method applies logistic regression to define the boundary of growth. The combination of these two different techniques can be a useful tool to handle the problem of extrapolation of predictive models at the growth limits.
... The CurveExpert ® program determined that the correlation coefficient between the model and experimental data is 0,97. For this article's purpose, this value is regarded as a valid correlation (Ratkowsky, 2002). With this, an analysis coefficient equation was obtained with the following equation: ...
Steam sterilization is a technique widely implemented in different biotechnological processes-among them the growth of microorganisms in bioreactors-, which require initial aseptic conditions. In the present research, the relationship between lethality and death rate of Bacillus cereus grown in 7-liter stirred-tank jacketed bioreactors was investigated, with sugar cane molasses as a carbon source. The sterilization process was tested with an industrial autoclave within data loggers in bioreactors, which measure the temperature in the cold point to quantify the accumulated lethality. Through data analysis, a contour plot allows a graphical prediction of the microbiological sterilization results in terms of colony forming units (CFU). 64 MONTERO-ZAMORA, MATA, MOLINA Y AGUILAR: Effect of Temperature-Time on... The study shows that there is a range between 16 and 20 minutes, approaching 123 °C, with a null presence of contaminant microorganisms. The surface chart demonstrates the existing non-linear relationship between the variables temperature and time involved. A positive correlation was observed using a Ratkowsky mathematical model with a 0,971 correlation coefficient and the estimated value of α = 11,04 and β =-1,49 for the non-linear model of parameterization. With these results, it is possible to predict the CFU based on F 0 data. This could provide an interesting basis for future sterilization practices and a methodology as a starting point for sterilization trials in industry and save time and costs. Resumen La esterilización con vapor húmedo es una técnica ampliamente utilizada en diferentes procesos biotecnológicos, entre ellos, el crecimiento de microorganismos en biorreactor, proceso que requiere condiciones asépticas. En la presente investigación, se estudió la relación entre letalidad y tasa de muerte del Bacillus cereus en un biorreactor enchaquetado de 7 litros, utilizando melaza como fuente de carbono. El proceso de esterilización fue realizado en una autoclave industrial con registros de temperatura en tiempo real colocados dentro del punto frío del biorreactor con el fin de cuantificar la letalidad acumulada. El análisis de los datos experimentales dio como resultado un gráfico de contorno que permite predecir el resultado de la esterilización en términos de unidades formadoras de colonias (UFC). El caso en estudio muestra que existe un intervalo entre los 16-20 minutos a una temperatura cercana a los 123 °C donde es posible eliminar en su totalidad el microorganismo contaminante. La superficie de respuesta demuestra que existe una relación no lineal entre las variables tiempo y temperatura, involucradas en el tratamiento térmico. Se observó una correlación positiva al utilizar el modelo matemático no lineal de Ratkowsky con un coeficiente de 0,971 y valores de α = 11,04 y β =-1,49 para la parametrización del modelo no lineal. Con los resultados obtenidos es posible predecir el valor de UFC presentes en el medio de cultivo para un valor determinado de F 0 , lo cual puede ser de amplio interés en procesos de esterilización industrial, con el fin de ahorrar tiempo y costos de operación. Palabras clave: Bioingeniería, fermentación, autoclavar, transferencia de calor, aséptico.
... The logistic type models is traditionally very often used in biology and in particular in microbiology for the description dynamic of microbial growth, infection processes and risk assessment (Peleg, 1997;Coleman, Marks, 1998;Ratkowsky, 2002;Pouillot, 2003). The simple continuous logistic model for microbial growth can be written as a first-order non-linear differential equation, that is ...
... Recently, Psomas, Nychas, Haroutounian, Panagiotis, and Skandamis (2011) Baranyi and Roberts (1994) primary model, coupled to a secondary temperature model, in order to simulate growth of a given microorganism during storage of a specific food or food category. A website has been developed (Psomas & Skandamis, 2010) in order to provide the users with free access to the software. The latter can be demonstrated with a PowerPoint presentation and it can be downloaded for free from http://www.aua.gr/psomas. ...
“Hurdle technology” is a worldwide technique of food preservation based on the application of a combination of generally mild treatments which act as “obstacles” that microflora must overcome to start to grow. Then, bacteria invest their energy in trying to maintain their homeostatic equilibrium instead of multiplying. While the action mechanisms underlying these treatments are not fully understood, it is very useful to know their effect on bacteria cells as well as the extension of such effects. Growth/no growth models have been developed to offer a response to this need. A review on growth/no growth microbial modeling is presented in this paper, addressing the most important factors and approaches employed. Five growth/no growth models of Salmonella published in the period 2001–2011 are reviewed, and their boundary regions were represented (Temperature vs pH) at two water activity values (0.983 and 0.990) and two cut-off probabilities (0.1 and 0.5). With this illustration, a picture of the relative grade of conservatism of the five models is provided. Additionally, the most important predictive tools in food microbiology (or tertiary models), including a software for growth/no growth modeling (Microbial Responses Viewer), are commented. Finally, some caveats in growth/no growth Salmonella modeling are addressed for future research.
... Logistic models. The logistic-type model is traditionally very often used in biology and in particular in microbiology for the description dynamic of microbial growth, infection processes and risk assessment (Peleg, 1997;Coleman and Marks, 1998;Ratkowsky, 2002;Pouillot et al., 2003). The simple continuous logistic model for microbial growth can be written as a first-order nonlinear differential equation, that is ...
... Two inactivation models were used to fit the experimental data of ultrasonic inactivation obtained by plate count, qPCR and PMA-qPCR methods: log-linear model (Bigelow and Esty, 1920) and Weibull model (Mafart et al., 2002). The statistics RMSE, R 2 and adjusted R 2 , all with a significant level of P 0.05, were used to determine the fitting of the data obtained by the different quantification methods, where a RMSE closer to 0 indicates a better fit (Ratkowsky, 2002) (Table 3). Plate count data from ultrasonic inactivation of E. coli O157:H7 followed a log-linear kinetic pattern (RMSE ¼ 0.0934; adjusted R 2 ¼ 0.9975) although the Weibull model also presented a very good fit to data (RMSE ¼ 0.1031; adjusted R 2 ¼ 0.9970). ...
The efficacy of sanitizing technologies in produce or in vegetable wash water is generally evaluated by plate count in selective media. This procedure is time consuming and can lead to misinterpretations because environmental conditions and sanitizing processes may affect bacterial growth or culturable capability. Thus, the aim of this study was to determine the applicability of a propidium monoazide real-time PCR (PMA-qPCR) method to monitor the inactivation by ultrasound treatment of foodborne bacteria in fresh-cut vegetable wash water. To this aim, lettuce wash water was artificially inoculated with Escherichia coli O157:H7 (10⁶ CFU/mL) and treated by means of a continuous ultrasonic irradiation with a power density of 0.280 kW/L. Quantification data obtained by PMA-qPCR and plate counts were statistically similar during the viability reduction of 99.996% which corresponds to 4.4 log reductions. Further reductions of E. coli O157:H7 were not detected by the PMA-qPCR method due to the limit of detection of this technique (20 CFU/mL). Inactivation data obtained by both techniques successfully fitted a linear model, giving no significant differences in kinetic parameters. These results indicate that the PMA-qPCR method is a suitable technique for evaluating ultrasonic disinfection of vegetable wash water, being able to distinguish between live and dead bacteria.
... Mean counts of E. coli O157:H7 were plotted versus contact time in Excel (Microsoft Corporation) spreadsheet and then were analyzed by non-linear regression to assess different inactivation kinetic models (Table 3). The statistics RMSE, R 2 and adjusted R 2 , all with a significant level of P < 0.05, were used to determine the best model for fitting where an RMSE closer to 0 indicates a better fit (Ratkowsky, 2002). All models were fitted to data by using the curve fitting toolbox provided by the GInaFiT (Geeraerd et al., 2006a(Geeraerd et al., , 2006b). ...
The efficacy of an electrochemical treatment in water disinfection, using boron-doped diamond electrodes, was studied and its suitability for the fresh-cut produce industry analyzed. Tap water (TW), and tap water supplemented with NaCl (NaClW) containing different levels of organic matter (Chemical Oxygen Demand (COD) around 60, 300, 550 ± 50 and 750 ± 50 mg/L) obtained from lettuce, were inoculated with a cocktail of Escherichia coli O157:H7 at 10⁵ cfu/mL. Changes in levels of E. coli O157:H7, free, combined and total chlorine, pH, oxidation-reduction potential, COD and temperature were monitored during the treatments. In NaClW, free chlorine was produced more rapidly than in TW and, as a consequence, reductions of 5 log units of E. coli O157:H7 were achieved faster (0.17, 4, 15 and 24 min for water with 60, 300, 500 and 750 mg/L of COD, respectively) than in TW alone (0.9, 25, 60 min and 90 min for water with 60, 300, 600 and 800 mg/L of COD, respectively). Nonetheless, the equipment showed potential for water disinfection and organic matter reduction even without adding NaCl. Additionally, different mathematical models were assessed to account for microbial inactivation curves obtained from the electrochemical treatments.
... Examples of simulated k(T) vs. T plots with Eq. (21) as a model are given in Fig. 16. Their appearance is almost identical to the curves shown by Ross and Dalgaard (2004), Ratkowsky (2004b), Gibson et al. (1994), Sautour et al. (2001) and others, when plotted either as k(T) vs. T or √ k(T) vs. T relationships. Notice that in contrast with an Arrhenius based model, the "growth term" in Eq. ...
Most of the models of microbial growth in food are Empirical algebraic, of which the Gompertz model is the most notable, Rate equations, mostly variants of the Verhulst's logistic model, or Population Dynamics models, which can be deterministic and continuous or stochastic and discrete. The models of the first two kinds only address net growth and hence cannot account for cell mortality that can occur at any phase of the growth. Almost invariably, several alternative models of all three types can describe the same set of experimental growth data. This lack of uniqueness is by itself a reason to question any mechanistic interpretation of growth parameters obtained by curve fitting alone. As argued, all the variants of the Verhulst's model, including the Baranyi-Roberts model, are empirical phenomenological models in a rate equation form. None provides any mechanistic insight or has inherent advantage over the others. In principle, models of all three kinds can predict non-isothermal growth patterns from isothermal data. Thus a modeler should choose the simplest and most convenient model for this purpose. There is no reason to assume that the dependence of the "maximum specific growth rate" on temperature, pH, water activity, or other factors follows the original or modified versions of the Arrhenius model, as the success of Ratkowsky's square root model testifies. Most sigmoid isothermal growth curves require three adjustable parameters for their mathematical description and growth curves showing a peak at least four. Although frequently observed, there is no theoretical reason that these growth parameters should always rise and fall in unison in response to changes in external conditions. Thus quantifying the effect of an environmental factor on microbial growth require that all the growth parameters are addressed, not just the "maximum specific growth rate." Different methods to determine the "lag time" often yield different values, demonstrating that it is a poorly defined growth parameter. The combined effect of several factors, such as temperature and pH or aw, need not be "multiplicative" and therefore ought to be revealed experimentally. This might not be always feasible, but keeping the notion in mind will eliminate theoretical assumptions that are hard to confirm. Modern mathematical software allows to model growing or dying microbial populations where cell division and mortality occur simultaneously and can be used to explain how different growth patterns emerge. But at least in the near future, practical problems, like translating a varying temperature into a corresponding microbial growth curve, will be solved with empirical rate models, which despite not being "mechanistic" are perfectly suitable for this purpose.
... The logistic type models is traditionally very often used in biology and in particular in microbiology for the description dynamic of microbial growth, infection processes and risk assessment (Peleg, 1997;Coleman, Marks, 1998;Ratkowsky, 2002;Pouillot, 2003). The simple continuous logistic model for microbial growth can be written as a first-order non-linear differential equation, that is ...
... A further development of the stochastic dynamical model proposed in this paper could also consist of using a growth/no-growth term [29,65,66]. This implies that fluctuating environmental conditions could cause the growth rate to go below a given threshold (no-growth region), contributing to the generation of a richer dynamics: both growth and no-growth cells of a bacterial population could have a non-vanishing probability to appear at the same time. ...
... This fact occurs because of the small number of replicates (n) in comparison to the number of conditions tested. In this situation, difficulties in the application of logistic regression models are found, especially when achieving convergence to a global optimum or to an appropriate set of conditions (Ratkowsky, 2002), unless alternative procedures are used (Geeraerd et al., 2004). By increasing the number of replicates (n = 30), smoother transitions between growth and no growth zones can be achieved. ...
The microbial behaviour of five enterotoxigenic strains of Staphylococcus aureus was studied in the growth/no growth domain. A polynomial logistic regression equation was fitted using a stepwise method to study the interaction of temperature (8, 10, 13, 16 and 19 degrees C), pH (4.5; 5.0; 5.5; 6.0; 6.5 7.0 and 7.5) and water activity (A(w)) (19 levels ranging from 0.867 to 0.999) on the probability of growth. Out of the 284 conditions tested, 146 were chosen for model data and 138 intermediate conditions for validation data. A growth/no growth transition was obtained by increasing the number of replicates per condition (n=30) in comparison to other published studies. The logistic regression model showed a good performance since 96.6% (141 out of 146 conditions) of the conditions for model data and 92.0% (127 out of 138 conditions) for validation data were correctly classified. The predictions indicated an abrupt growth/no growth interfaces occurred at low levels of temperature, pH and A(w). At 8 degrees C, S. aureus grew only at optimum levels of pH and A(w) while at temperatures above 13 degrees C, growth of S. aureus was observed at pH=4.5 and A(w)=0.96 (13 degrees C), 0.941 (16 degrees C) and 0.915 (19 degrees C). The optimal pH at which growth of S. aureus was detected earlier was 6.5. However, a slight decrease of the probability of growth was noticed in the pH interval of 7.0-7.5 at more stringent conditions. The ability of S. aureus to grow at low A(w) was shown since growth was detected at A(w)=0.867 (T=19 degrees C; pH=7.0). Finally, a comparison of model predictions with literature data on growth/no growth responses of S. aureus in culture media and cooked meat was made. Model predictions agreed with published data in 94% of growth cases and in 62% of no growth cases. The latter discordance is highly associated to other environmental factors (such as other preservatives, strains etc.) included in published models that did not match the ones included in our study. This study can help manufacturers in making decision on the most appropriate formulations for food products in order to prevent S. aureus growth and enterotoxin production along their shelf-life.
... Later, Salter et al. (39) used the nonlinear logistic regression, which allows estimation of all parameters of the model. The advantages and problems of nonlinear logistic regression in predictive microbiology are extensively discussed in a publication by Ratkowsky (35). ...
The growth limits of a mixture of five strains of Salmonella Typhimurium in tryptic soy broth were examined at different environmental conditions. The response of the pathogen was monitored in a total of 350 combination treatments of temperature (10 to 35 degrees C), pH (3.76 to 6.44), and water activity (aw, 0.913 to 0.990) for 62 days. No growth/growth (turbidity) data were modeled by logistic polynomial regression. The concordance index of the logistic model was 99.8%, indicating a good fit to the observed data. The minimum pH and aw values that permitted growth were 3.94 and 0.942, respectively, and occurred in the temperature range of 25 to 35 degrees C. At temperatures below this range, the minimum pH and aw allowing growth increased as the temperature decreased. The results showed an abrupt change in the probability of growth close to the boundary with minor changes of the environmental factors. The probabilities predicted by the model were compared with published data on the actual response of Salmonella Typhimurium or other salmonellae serotypes in 50 cases of food products, including salad dressing, mayonnaise, meat, cheese, vegetables, and fruits. The model predicted successfully the response of the pathogen in 90% of the tested cases. The results of the study indicated that the developed model predicts satisfactorily the growth/no growth interface of Salmonella Typhimurium in foods and can provide useful quantitative data for the development of safer food products and processes.
... The observed survival ratio is Ŝ t ϭ N t /N 0 , and N t is the observed remaining population. Besides the SSE, the most simple and informative goodness measure of fit for regression models, both linear and nonlinear, is the root mean square error (RMSE) (24,25), given by equation 4: ...
Fresh vegetables contaminated with Yersinia enterocolitica have been implicated in foodborne disease outbreaks. Surfaces of vegetables can become contaminated with pathogenic microorganisms through contact with soil, irrigation water, fertilizers, equipment, humans, and animals. One approach to reduce this contamination is to treat fresh produce with sanitizers. In this study, the ability of ozone to inactivate Y. enterocolitica inoculated in water and on potato surfaces was evaluated. Furthermore, the efficacy of ozone in reducing natural flora on whole potato was determined. Total aerobic mesophilic and psychrotrophic bacteria, total coliforms, and Listeria monocytogenes were enumerated. Finally, several disinfection kinetic models were considered to predict Y. enterocolitica inactivation with ozone. Treatments with ozone (1.4 and 1.9 ppm) for 1 min decreased the Y. enterocolitica population in water by 4.6 and 6.2 log CFU ml(-1), respectively. Furthermore, ozonated water (5 ppm) for 1 min decreased Y. enterocolitica and L. monocytogenes from potato surfaces by 1.6 and 0.8 log CFU g(-1), respectively. Therefore, ozone can be an effective treatment for disinfection of wash water and for reduction of potato surface contamination.
... For the models described, a simple hierarchy construct is useful: the simplest is nominal logistic, followed by response surface, followed by mechanistic or truly predictive models. If the purpose of the model is to describe under which conditions a likely product will fail, then a nominal logistic approach is very useful (31). It gives the simple criterion of growth or no growth. ...
The Gamma hypothesis, that multiple inhibitory factors combine independently, is the underlying hypothesis for the quantification of the Hurdle concept used in food manufacture. The literature, however, is confused as to whether interactive effects exist and under which circumstances they occur, if at all. Using the method of time to detection (TTD), the inhibitory effect of pH, salt and specific weak acids (acetic, propionic, sorbic and benzoic) and combinations of these with respect to the growth of Aeromonas hydrophila (ATCC 7966) were analysed. A model based on the relative rate to detection described all combinations analysed as having independent effects on the TTD. No synergistic interactions were found between pH and salt, between pH and individual weak acids or between combinations of weak acids and pH for any of the systems under study. This study supports the validity of the Gamma concept -- that individual environmental effects act independently and should, in turn, facilitate attempts to model the growth of other microorganisms under a variety of conditions.
... This assumption was implicit while using the following transformation ffiffiffiffiffiffiffiffiffiffiffiffiffiffi P þ 50 p (Pardo et al., 2005a,b) or explicit while using the logit transformation Ln[P / (100 − P)] (Huang et al., 2001; Perryman et al., 2002; Kalolewski et al., 2004 ). In predictive microbiology , the logit transformation was used for modelling the growth/ no growth interface that involved the use of probability models, where the response variable was typically binomial (Ratkowsky, 2002). When kinetic models are used such as in this study, the logit transformation should be avoided. ...
The objectives of this study were i/ to examine germination data sets over a range of environmental conditions (water activity, temperature) for eight food spoilage moulds, ii/ to compare the ability of the Gompertz equation and logistic function to fit the experimental plots, iii/ to simulate germination by assessing various distributions of the latent period for germination amongst a population of spores. Data sets (percentage germination, P (%), versus time, t) of Aspergillus carbonarius, Aspergillus ochraceus, Fusarium verticillioides, Fusarium proliferatum, Gibberella zeae, Mucor racemosus, Penicillium chrysogenum and Penicillium verrucosum were analysed. No correlation, or relationship between the mean percentage [mean (P)] and the variance [var (P)] was found. Therefore no transformation of the germination data was required. Experimental data were fitted by using the Gompertz equation P = A exp (-exp [mu(m) e/A (delta - t) + 1]) and the logistic function P = Pmax/(1 + exp (k (tau - t))). Based on the residual mean square error (RMSE), no model performed better than the other one. However, model parameters were generally determined more precisely with the logistic model than with the Gompertz one. The time course of fungal spore germination curves was simulated assuming different distributions of the latent period for germination, lag, amongst a population of spores. The growth rate of germ tubes was calculated by means of the relationship: lag x rate = k. For normal Gaussian distributions, germination curves were symmetrical with respect to the inflection point and should be modelled with the logistic function. Skewed distributions were capable of simulating an asymmetric germination curve that was fitted by the Gompertz model. Future studies should be conducted for assessing whether the distributions assumed in this paper are in accordance with the experimental distributions that are still unknown.
... Regarding nonlinear regression analysis, the fitting process was performed by minimizing the sum of squares of the errors (SSE) and root mean square error (RMSE) (Ratkowsky, 1983;Ratkowsky, 2002) where a RMSE closer to 0 indicates a better fit. The statistics SSE and RMSE all with a significant level of Po0.05, have been used to determine the best model for fitting the experimental reductions of S. sonnei in ozonated water with initial ozone concentrations of 1.6 and 2.2 ppm. ...
Several outbreaks of shigellosis have been attributed to the consumption of contaminated fresh-cut vegetables. The minimal processing of these products make it difficult to ensure that fresh produce is safe for consumer. Chlorine-based agents have been often used to sanitize produce and reduce microbial populations in water applied during processing operations. However, the limited efficacy of chlorine-based agents and the production of chlorinated organic compounds with potential carcinogenic action have created the need to investigate the effectiveness of new decontamination techniques. In this study, the ability of ozone to inactivate S. sonnei inoculated on shredded lettuce and in water was evaluated. Furthermore, several disinfection kinetic models were considered to predict S. sonnei inactivation with ozone. Treatments with ozone (1.6 and 2.2 ppm) for 1 min decreased S. sonnei population in water by 3.7 and 5.6 log cfu mL(-1), respectively. Additionally, it was found that S. sonnei growth in nutrient broth was affected by ozone treatments. After 5.4 ppm ozone dose, lag-phases were longer for injured cells recovered at 10 degrees C than 37 degrees C. Furthermore, treated cells recovered in nutrient broth at 10 degrees C were unable to grow after 16.5 ppm ozone dose. Finally, after 5 min, S. sonnei counts were reduced by 0.9 and 1.4 log units in those shredded lettuce samples washed with 2 ppm of ozonated water with or without UV-C activation, respectively. In addition, S. sonnei counts were reduced by 1.8 log units in lettuce treated with 5 ppm for 5 min. Therefore, ozone can be an alternative treatment to chlorine for disinfection of wash water and for reduction of microbial population on fresh produce due to it decomposes to nontoxic products.
... Therefore, by modifying the environmental factors, formulations of minimally processed food products can be designed in order to increase food safety regarding L. monocytogenes growth. Ratkowsky (2002) noticed certain problems that arise from the use of linear and non linear logistic regression models. The use of small numbers of replicates per combination of environmental factors (due to the timeconsuming nature of the experimental work) does not allow convergence to a global optimum or an appropriate set of conditions. ...
A new approach to predict the growth/no growth interface of Listeria monocytogenes as a function of storage temperature, pH, citric acid (CA) and ascorbic acid (AA) is presented. A linear logistic regression procedure was performed and a non-linear model was obtained by adding new variables by means of a Neural Network model based on Product Units (PUNN). The classification efficiency of the training data set and the generalization data of the new Logistic Regression PUNN model (LRPU) were compared with Linear Logistic Regression (LLR) and Polynomial Logistic Regression (PLR) models. 92% of the total cases from the LRPU model were correctly classified, an improvement on the percentage obtained using the PLR model (90%) and significantly higher than the results obtained with the LLR model, 80%. On the other hand predictions of LRPU were closer to data observed which permits to design proper formulations in minimally processed foods. This novel methodology can be applied to predictive microbiology for describing growth/no growth interface of food-borne microorganisms such as L. monocytogenes. The optimal balance is trying to find models with an acceptable interpretation capacity and with good ability to fit the data on the boundaries of variable range. The results obtained conclude that these kinds of models might well be very a valuable tool for mathematical modeling.
... The latter model type has the advantage to contain biological meaningful parameters. These parameters, however, appear in a non-linear way hampering parameter estimation (Ratkowsky, 2002). Fixing these parameters yields a linear regression problem but may lead to a suboptimal model if the fixed values are erroneous. ...
Growth/no growth models were developed for two spoilage bacteria typical for acidified sauces, L. plantarum and L. fructivorans. Influencing factors embedded in the model are also those typically encountered in these acidified sauces. The pH was varied between 3.0 and 5.0 (5 levels), and the acetic and lactic acid concentration ranged from 0 to 3% (6 levels). Modified MRS broth was inoculated at a high inoculation level (10(6) CFU/ml), incubated at 30 degrees C and growth was assessed by optical density measurements. All combinations of environmental conditions were tested in twelvefold yielding precise values for the probability of growth. Data were modelled by means of ordinary logistic regression. A comparison was made between a model containing the total acid concentrations as explanatory variables, on the one hand, and a model differentiating between the dissociated and undissociated concentrations, on the other hand. Results showed that (i) L. plantarum and L. fructivorans behave differently, resulting in a clearly distinct growth/no growth interface, (ii) there was no great difference between the established models with different explanatory variables, (iii) in some cases, growth/no growth boundaries at very low probabilities (which are more practical in industry) show illogical behaviour. The results of this study were also compared with the CIMSCEE code, which is often used by food producers to determine the stability of their acidified food products.
Logistic regression models quantify the occurrence of microbial development in a real scenario from production and commercialization to consumption of the product. Recent studies have demonstrated the versatility of its application as a tool to achieve microbiological food quality and safety. However, more knowledge of the methodology to be followed is needed, as well as the computer software available to be used. This chapter describes the mathematical principles, procedure, and validation to develop a growth/no-growth model and its possible applications in the various areas of food science.Key wordsProbabilistic modelLogistic regressionSpoilage microorganismsPathogen microorganismsFood qualityMathematical model
This study aimed to evaluate the survival of Listeria monocytogenes on organic Honey Crisp (HC) and Fuji (FJ) apples during storage at various temperatures. Fresh organic HC and FJ apples (without waxing coating) obtained from local wholesale market were inoculated with a 2-strain mixer of L. monocytogenes followed by storing at 5.0 [22.9% Relative Humidity (RH)], 12.0 (37.0% RH), and 22.5 °C (50.4% RH) for 60, 35, 7 days, respectively, and periodically (day-0 to 60) analyzing microbial populations. Surviving L. monocytogenes were spread-plated on Modified-Oxford agar after 10 or 100-fold serial dilution. Data was analyzed using the mixed-model-procedure of SAS and GinaFit software. The initial populations of L. monocytogenes on HC and FJ apples were 6.23–6.89 log10CFU/apple for storage at 5, 12, and 22.5 °C. The pathogen survival cell counts decreased (P < 0.05) to 2.34 to 4.05, 2.72 to 2.98, and 2.47 to 3.75 log10CFU/apple by the end of the storage at 5, 12, and 22.5 °C. L. monocytogenes was more vulnerable (P < 0.05) on FJ than HC apples and at room temperature than cold storage temperatures. The inactivation parameters calculated from the Linear, Weibull, and Biphasic models generally are consistent with the pathogen survival curves with few exceptions. Results of this study filled the data gap for understanding of microbiological risks associated with postharvest practices of tree fruit production. Future studies are needed to quantify the natural wax amount on various organic apples and develop pre- and postharvest intenvention strategies for inactivation of foodborne pathogens on apples as well as other tree fruits.
The objective of this study was to determine the probabilities of spoilage of different formulations of cakes varying pH, preservatives, and water activity (aw) at different storage temperatures by Penicillium citrinum LMQA_053. Cake were produced with different aw values (0.78, 0.80, 0.83 and 0.86), pH (6.0; 6.3; 6.6 and 7.0), calcium propionate (0%, 0.10%, 0.15% and 0.20%) and potassium sorbate (0%, 0.05% and 0.10%). Then, cakes were inoculated with P. citrinum LMQA_053 and stored at 20, 25, and 30 °C. The logistic regression model was adjusted to growth/no-growth results to obtain predictions of growth probability. The probabilistic model presented 83% of agreement with the observed results. A chocolate cake formulation combining low aw (<0.78) and the highest concentration of preservatives (0.1% potassium sorbate, 0.2% calcium propionate) could prevent the growth of P. citrinum LMQA-053 in all pH and temperatures assessed for 45 days. The developed model was able to predict the growth probability of P. citrinum LMQA_053 and can be used to develop more robust formulas towards avoiding mould growth and spoilage of cakes.
The growth/no growth boundary model of Bacillus simplex was developed using logistic regression and neural network as a function of pH, heating temperature, and water activity (aw). The model was based on the bacterial responses in tryptic soy broth (TSB) with an initial bacterial count of ca. 10 CFU/mL, which was in turn dependent on 192 conditions comprising four levels each of pH (7.0, 6.6, 6.2, and 5.8), heating temperature (70, 80, 85, and 90 °C for each 10 min), and storage period (1, 2, 3, and 4 weeks), and three levels of aw (0.97, 0.98, and 0.99). To evaluate definite growth probability, 60 repetitive experiments were performed per condition, which resulted in total 11,520 datasets. The developed models were evaluated using independent experimental dataset of growth/no growth in TSB and carbonara sauce, an example of nutrient-rich processed food matrix. Both developed models accurately described the growth/no growth boundary of B. simplex. Although all the four parameters significantly influenced the growth/no growth boundary, the heating temperature and storage period exhibited greater effect than pH and aw parameters under the examined conditions. The fraction correct (FC) values of independent verification data in TSB for the logistic regression model and the neural network model were 92.2% and 89.8%, respectively. Developing a model with three replicated experiments, which are conventionally and frequently used in the previous studies, may lead to incorrect judgement of growth/no growth. Models based on such inaccurate datasets result in inaccurate prediction. For example, the area under the receiver operating curve (AUC), an index of the model accuracy, showed lower value. Based on the relationship between the replicates of the experiments and AUC, the minimum requirements of the replicated experiment was found to be ≥ 10 and ≥12 times for the logistic regression model and neural network model, respectively. The results clearly demonstrated the minimum requirements of experimental replicates for the development of stochastically definite growth/no growth boundary model.
This study aims to assess how small produce growers handle containers and evaluate the survival of Salmonella and Listeria monocytogenes on various produce container surfaces commonly used at farmers markets, under storage conditions both at refrigerated and room temperature. In Study I, an anonymous survey was conducted to assess the practices of handling produce containers from 28 vendors at farmers markets in Morgantown, WV and 141 vendors from farmers markets in Kentucky. In Study II, plastic, pressed-card, and wood containers were trimmed (25 cm²) and inoculated with S. typhimurium and Tennessee, and two strains of L. monocytogenes, stored at 3.2 °C (22.19% RH) and 22.5 °C (50.40% RH), respectively, for 21 days and periodically analyzed for microbial populations on XLT-4 (Salmonella) and Modified-Oxford (L. monocytogenes) agars. The survey results showed that plastic, paper, and wood containers are the top three choices for small produce growers to transport and present produce at farmers markets. The pathogens decreased slower (P < 0.05) at 3.2 °C and on pressed-card and wood surfaces than at 22.5 °C and on a plastic surface. At 3.2 °C, Salmonella counts decreased (P < 0.05) from 5.27 to 5.53 to 2.63–2.84 log CFU/cm², and L. monocytogenes decreased (P < 0.05) from 6.39 to 6.93 to 4.89–5.46 log CFU/cm² on the three material surfaces by the end of the storage period, with the lowest (P < 0.05) survival on a plastic surface. At 22.5 °C, Salmonella populations decreased (P < 0.05) from 4.94 to 5.38 to <1.30 log CFU/cm² (the detection limit) after 3, 9 and 12 days on plastic, pressed-card and wood surfaces, respectively. L. monocytogenes decreased (P < 0.05) from 6.39 to 6.93 to ≤1.30 log CFU/cm² after 12, 12, and 21 days on plastic, wood and pressed-card surfaces, respectively. These results were confirmed by different mathematical survival models for analyzing pathogen inactivation rates. Vendors at farmers markets should choose plastic containers to store fresh produce and avoid storing containers at refrigerated temperature.
Predictive microbiology is a relatively novel scientific field belonging to food microbiology; it is aimed at developing mathematical models that account for the effect of intrinsic and extrinsic factors on microorganism responses in food. This area has experienced significant advances over the last decades, providing numerous and innovative predictive models to estimate the fate of microorganisms across the food chain. Predictive models can be classified, according to the type of modelled phenomenon, into growth models, inactivation and survival models and probability models. More recently, transfer models, single-cell-based models or genomic-scale models have been proposed as more mechanistic approaches to reflecting microbial responses in foods. Predictive microbiology is, in addition, the basis for developing quantitative microbial risk assessment by modelling different microbial process across the food chain, and thus supporting making decision processes within risk management. Likewise, the application of user-friendly software tools has enabled end-users to easily apply predictive models in different contexts (e.g. Pathogen Modeling Program ? PMP, ComBase or Microhibro). In spite of these important achievements, predictive microbiology should still continue to improve model precision and accuracy and, based on more mechanistic approaches such as systems biology, to generate robust and reliable models for complex and real food systems.
Predictive models have been initially focused on the estimation of kinetic parameters, as described in the preceding chapter. However, other modeling approaches are often requested, especially when considering the transmission of a pathogen along the food chain or the probability that this pathogen can grow or survive at certain environmental conditions. This is the underlying reason why transfer and growth/no growth models presented a relevant development in predictive microbiology. These models can be effectively applied when presence/absence data are required, or in specific food processes. Alternatively, survival and transmission of microorganisms through food contact surfaces, environment and between different foods can be also estimated. Additional advantages, such as the extent of shelf life, or the effect of novel preservation methods in minimally processed foods provide a wider application of predictive microbiology. Bacterial transfer models and growth/no growth models are described in this chapter.
Introduction Classification of models Description of main models Applications of predictive microbial modelling Predictive microbial modelling and quantitative risk assessment Conclusions
Obtaining data for improving food safety management systems is often required to assist decision making in a short timeframe, potentially allowing decisions to be made and practices to be implemented in real time. Collection, storage, and retrieval of new data regarding microbial responses in foods gain insight on the achievement of food safety management measures (i.e., food safety objectives, performance objectives), avoiding the increase of fail-dangerous events. The role of data bases in predictive microbiology has been widely demonstrated as useful tools for the development of computing software or tertiary models, which allow users to estimate growth, survival, or inactivation of food-borne pathogens and spoilage microorganisms in different food matrices. Additionally, the fast development of information and communication technologies (ICTs) has increased the software tools available in predictive microbiology. These tools, named tertiary models, are created for a wide range of applications and types of users: scientists, food operators, risk managers, etc. Although early versions were designed as standalone systems, nowadays on-line software is a major trend making tools available everywhere to everyone through the Internet. In this chapter, descriptive examples of data bases and software tools used in predictive microbiology are explained.
This chapter presents that meat products are perishable and unless processed, packaged, distributed, and stored appropriately can spoil in relatively short time. Over-growth of incidental pathogenic bacteria like Listeria monocytogenes, Salmonella sp. and diarrheagenic Escherichia coli followed by under-cooking or inadequate preparation may pause a potential hazard for the consumer. Despite the proliferation of food safety regulations and the introduction of safety management systems, such as Hazard Analysis Critical Control Point (HACCP), risk assessment studies show that foodborne disease has remained a main concern in the last decade. The chill chain itself is the weakest segment of quality and safety assurance systems for chilled foods, with temperature frequently deviating from specifications. Efforts should be directed toward developing an effective chill chain management system that optimizes quality distribution and minimizes risk at the time of consumption. The goal would be to replace the conventional First in First out (FIFO) approach with a new system, based on actual risk evaluation at important points in the chill chain, through continuous product temperature monitoring and quality data input.
In wastewater treatment, the rate of ammonia oxidation decreases with pH and stops very often slightly below a pH of 6. Free ammonia (NH3) limitation, inhibition by nitrous acid (HNO2), limitation by inorganic carbon or direct effect of high proton concentrations have been proposed to cause the rate decrease with pH as well as the cessation of ammonia oxidation. In this study, we compare an exponential pH term common for food microbiology with conventionally applied rate laws based on Monod-type kinetics for NH3 limitation and non-competitive HNO2 inhibition as well as sigmoidal pH functions to model the low pH limit of ammonia oxidizing bacteria (AOB). For this purpose we conducted well controlled batch experiments which were then simulated with a computer model. The results showed that kinetics based on NH3 limitation and HNO2 inhibition can explain the rate decrease of ammonia oxidation between pH 7 and 6, but fail in predicting the pH limit of Nitrosomonas eutropha at pH 5.4 and rates close to that limit. This is where the exponential pH term becomes important: this term decreases the rate of ammonia oxidation to zero, as the pH limit approaches. Previously proposed sigmoidal pH functions that affect large pH regions, however, led to an overestimation of the pH effect and could therefore not be applied successfully. We show that the proposed exponential pH term can be explained quantitatively with thermodynamic principles: at low pH values, the energy available from the proton motive force is too small for the NADH production in Nitrosomonas eutropha and related AOB causing an energy limited state of the bacterial cell. Hence, energy limitation and not inhibition or limitation of enzymes is responsible for the cessation of the AOB activity at low pH values.
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This paper considers the applications of the European Commission (EC) Regulation N degrees 2073/2005 concerning Listeria monocytogenes in ready-to-eat (RTE) food products. A simplified modelling approach (SMA), which is intended for a practical evaluation of the behaviour of the pathogen during processing and storage of an RTE meat product (Pitina) from traditional producers, was presented. This approach included a growth/no-growth model, which was developed by using the limits adopted as safety criteria by the EU, a model based on the gamma concept (GM) and a kinetic three-phase model (TPL). Based on the findings of the present study, Pitina was incapable of supporting the growth of L monocytogenes and the no-growth conditions assessed through the shelf life by the model were confirmed by challenge testing. When the simplified approach was used to estimate the total pathogen growth relative to the Pitina multistage process, taking into account the effects of various hurdles, it provided predictions of L monocytogenes growth corresponding to the observed data in the inoculation studies. Therefore, this simplified approach is expected to enable the food producers to identify appropriate processing conditions. The successfully validated SMA was found sufficiently complex to predict growth responses of L monocytogenes in RTE foods, but at the same time easy-to-use for practical processing situations.
Predictive models for the heat inactivation of Listeria monocytogenes in pure and mixed culture biofilms, formed on stainless steel and rubber surfaces and in the presence of food soil were developed using fraction negative data and logistics regression. The validation study indicated that at the 50% probability level of L. monocytogenes inactivation, the predictive model with strain 3990 on stainless steel surfaces was conservative in its estimate of L. monocytogenes biofilm inactivation while the Scott A model was not a reliable predictor of the heat inactivation of L. monocytogenes in a biofilm. The multispecies (L. monocytogenes, Pseudomonas and P. agglomerans) biofilm was an adequate predictor of L. monocytogenes biofilm inactivation and can be used in situations of low occurrence in a food product. A predictive model for the heat inactivation of L. monocytogenes on rubber surfaces was developed. The model provides for three prediction situations in the presence of soil, the fairly conservative assessment of risk using heat resistant Scott A strain and the less conservative predictions based on strain 3990 and Listeria in a mixed culture. For the low soil condition, the Scott A and 3990 strains showed adequate assessment of heat inactivation while the L. monocytogenes in a multi-species biofilm was conservative in its predictions. These predictive models could be used as a guide to apply hot water sanitation when chemical sanitation is ineffective for a process. Heat stress induced the increased production of biofilm for L. monocytogenes Scott A. There were changes in the proteins expression before and after heat stress.
An alternative predictive model for microbial inactivation and a novel web-based tool for the application of predictive microbiology are reviewed in this paper. The developed model, based on probabilistic concepts, enabled the identification of minimum processing conditions necessary to obtain a required log reduction, regardless of the underlying inactivation kinetics. The model also provides the probability distribution of the inactivation effect. The revised web-tool, the MRV (Microbial Responses Viewer), provides information concerning growth/no growth boundary conditions and the specific growth rates of queried microorganisms. The MRV enables users to retrieve microbial growth/no growth information intuitively. Using the MRV, food processors can easily identify appropriate food design and processing conditions.
Predictive microbiology has been developed utilizing accurate and versatile mathematical models to predict microbial behavior during food processing instead of using traditional microbiological methods. Generally, mathematical models are classified into three types: primary, secondary, and tertiary models. Secondary model deals with the response of parameters appearing in primary modeling approaches as functions of one or more environmental conditions like temperature, pH, etc. Most of them have little or no microbiological basis, which makes interpretation of some model parameters difficult and sometimes their performance is not stable. This paper reviews the performance and development of secondary models and discusses their strengths and weaknesses, which may help future model development and application.
In predictive microbiology, mathematical models are developed that can describe the behavior of microorganisms given certain (dynamic) environmental conditions. Ultimately, these models enable the prediction of microbial behavior in real food products. Two mathematical techniques or approaches (i.e., design of experiments and optimal experiment design for parameter estimation) that can increase and/or optimize the information contained in a (series of) experiment(s) are discussed in this chapter. Hereafter, predictive models are presented in the following subdivisions: (1) kinetic models, (2) probabilistic models, and (3) dose-response models. After model structure selection, model parameters are to be estimated. Methods to quantify parameter estimation uncertainty are listed while different graphical and quantitative methods that exist to evaluate the model performance are discussed. This chapter overall describes the model-building process in a conceptual approach focusing on specific examples related to food decontamination processes.
Thermal processing or cooking of food products has been adopted for
centuries as a method of food preservation. Enhancement of product
quality parameters such as color, flavor, and texture probably
contributed to the adoption of the method for a variety of products.
Today, cooking or thermal processing is one of the most commonly used
unit operation in the food industry. The significant advantages to
cooking of meat and poultry products include extension of shelf life,
desirable organoleptic properties, enhanced economic value, and
assurance of safety of the products.
This book, inclusive of 22 chapters, provides a comprehensive discussion on the use of natural antimicrobials in the food industry for enhancing overall food safety and quality. In particular, specific chapters are devoted to the discussion of bacteriophages and bacterial antimicrobial compounds, algae-derived antimicrobials, fungi-derived secondary antimicrobial metabolites, antimicrobials derived from animal byproducts, and naturally-occurring antimicrobials derived from plants and plant products. Chapters discussing the reduction of biogenic amines in meat products, as well as an online database for natural antimicrobials, are also included. This book will be of valuable resource for professionals engaged in the food industry and academe, as well as for students of food-related courses.
The present paper discusses the use of modified Lotka-Volterra equations in order to stochastically simulate the behaviour of Listeria monocytogenes and Lactic Acid Bacteria (LAB) during the fermentation period (168 h) of a typical Sicilian salami. For this purpose, the differential equation system is set considering T, pH and aw as stochastic variables. Each of them is governed by dynamics that involve a deterministic linear decrease as a function of the time t and an "additive noise" term which instantaneously mimics the fluctuations of T, pH and aw. The choice of a suitable parameter accounting for the interaction of LAB on L. monocytogenes as well as the introduction of appropriate noise levels allows to match the observed data, both for the mean growth curves and for the probability distribution of L. monocytogenes concentration at 168 h.
Several model types have already been developed to describe the boundary between growth and no growth conditions. In this article two types were thoroughly studied and compared, namely (i) the ordinary (linear) logistic regression model, i.e., with a polynomial on the right-hand side of the model equation (type I) and (ii) the (nonlinear) logistic regression model derived from a square root-type kinetic model (type II). The examination was carried out on the basis of the data described in Vermeulen et al. [Vermeulen, A., Gysemans, K.P.M., Bernaerts, K., Geeraerd, A.H., Van Impe, J.F., Debevere, J., Devlieghere, F., 2006-this issue. Influence of pH, water activity and acetic acid concentration on Listeria monocytogenes at 7 degrees C: data collection for the development of a growth/no growth model. International Journal of Food Microbiology. .]. These data sets consist of growth/no growth data for Listeria monocytogenes as a function of water activity (0.960-0.990), pH (5.0-6.0) and acetic acid percentage (0-0.8% (w/w)), both for a monoculture and a mixed strain culture. Numerous replicates, namely twenty, were performed at closely spaced conditions. In this way detailed information was obtained about the position of the interface and the transition zone between growth and no growth. The main questions investigated were (i) which model type performs best on the monoculture and the mixed strain data, (ii) are there differences between the growth/no growth interfaces of monocultures and mixed strain cultures, (iii) which parameter estimation approach works best for the type II models, and (iv) how sensitive is the performance of these models to the values of their nonlinear-appearing parameters. The results showed that both type I and II models performed well on the monoculture data with respect to goodness-of-fit and predictive power. The type I models were, however, more sensitive to anomalous data points. The situation was different for the mixed strain culture. In that case, the type II models could not describe the curvature in the growth/no growth interface which was reversed to the typical curvatures found for monocultures. This unusual curvature may originate from the fact that (i) an interface of a mixed strain culture can result from the superposition of the interfaces of the individual strains, or that (ii) only a narrow range of the growth/no growth interface was studied (the local trend can be different from the trend over a wider range). It was also observed that the best type II models were obtained with the flexible nonlinear logistic regression, although reasonably good models were obtained with the less flexible linear logistic regression with the nonlinear-appearing parameters fixed at experimentally determined values. Finally, it was found that for some of the nonlinear-appearing parameters, deviations from their experimentally determined values did not influence the model fit. This was probably caused by the fact that only a limited part of the growth/no growth interface was studied.
The gamma hypothesis states that there are no interactions between antimicrobial environmental factors. The time to growth
of Aeromonas hydrophila challenged with pH, NaNO2, and salt combinations at 30°C was investigated. Data were examined using a model based on the gamma hypothesis (the gamma
model), which takes into account variance-stabilizing transformations and which gives biologically relevant parameters. At
high concentrations of NaNO2 and at pHs of >6.0, the antimicrobial action of the nitrite ion has a strong influence (MIC = 2,033 mg liter−1), whereas at pHs of <6, nitrous acid is dominant (MIC = 1.5 mg liter−1). This change is not due to a “synergy” between pH and the nitrite ion but is due to the shift in the equilibrium concentrations
of nitrous acid and nitrite in solution caused by pH. In combination with salt, the parameters found for the action of Na
nitrite were identical to those found when it was examined in isolation. Therefore, pH, NaNO2, and salt act independently on the growth of A. hydrophila. By expanding the gamma model with a cardinal temperature model, the results of fitting the model of Palumbo et al. (J. Food
Prot. 54:429-435, 1994) to randomly produced environmental conditions could be reproduced, suggesting that temperature also has an
independent effect.
The survival/death interface model was developed for prediction of inactivation of Listeria monocytogenes by high pressure processing (HPP). The model was derived from data sets comprising 360 combinations of environmental factors such as pressure (200, 300 400, and 500 MPa), pressure-holding time (1, 3, 5, 10, 20, 30 min), pH (3, 4, 5, 6, 7), and inoculum level (3, 5, 7 log(10) CFU/ml). The determination of survival/death of L. monocytogenes after HPP was confirmed by the presence/absence of colony forming ability on non-selective agar plates after 30 days of incubation at 20 degrees C in broth to take into account recovery of HPP-induced injured cells. The developed linear logistic model with time logarithmically transformed gave a degree of agreement between probabilities predicted by the fitted model and all observations as 99.3% concordant. The model provided a good fit to the data as shown by performance statistics. The developed interface model in the present study provided requisite process conditions for the target effect of HPP on L. monocytogenes. In addition to using the simple linear logistic model, a polynomial logistic model was also fitted to the data where pressure-holding time was not logarithmically transformed. That model did not produce a better fit to the data and resulted in some potentially misleading predictions. Optimization of HPP could be accomplished using the model developed in this study. Furthermore, choice in processing factors allows for processing flexibility in HPP and specifies the process criteria that are incorporated into the HACCP plan.
The form of a previously developed Bĕlehrádek type of growth rate model was used to develop a probability model for defining the growth/no growth interface as a function of temperature (10 to 37 degrees C), pH (pH 2.8 to 6.9), lactic acid concentration (0 to 500 mM), and water activity (0.955 to 0.999; NaCl was used as the humectant). Escherichia coli was unable to grow in broth in which the undissociated lactic acid concentration exceeded 11 mM or, with two exceptions, at a pH of 3.9 or less with no lactic acid present. Under experimental conditions at which the pH and the undissociated acid concentrations were the major growth-limiting factors, the growth/no growth interface was essentially independent of temperature at temperatures ranging from 15 to 37 degrees C. The interface between conditions that allowed growth and conditions at which growth did not occur was abrupt. The inhibitory effect of combinations of water activity and pH varied with temperature. Predictions of the model for the growth/no growth interface were consistent with 95% of the experimental data set.
Models describing the limits of growth of pathogens under multiple constraints will aid management of the safety of foods
which are sporadically contaminated with pathogens and for which subsequent growth of the pathogen would significantly increase
the risk of food-borne illness. We modeled the effects of temperature, water activity, pH, and lactic acid levels on the growth
of two strains ofListeria monocytogenes in tryptone soya yeast extract broth. The results could be divided unambiguously into “growth is possible” or “growth is
not possible” classes. We observed minor differences in growth characteristics of the two L. monocytogenes strains. The data follow a binomial probability distribution and may be modeled using logistic regression. The model used
is derived from a growth rate model in a manner similar to that described in a previously published work (K. A. Presser, T.
Ross, and D. A. Ratkowsky, Appl. Environ. Microbiol. 64:1773–1779, 1998). We used “nonlinear logistic regression” to estimate
the model parameters and developed a relatively simple model that describes our experimental data well. The fitted equations
also described well the growth limits of all strains of L. monocytogenesreported in the literature, except at temperatures beyond the limits of the experimental data used to develop the model (3
to 35°C). The models developed will improve the rigor of microbial food safety risk assessment and provide quantitative data
in a concise form for the development of safer food products and processes.
From the reviews of the First Edition."An interesting, useful, and well-written book on logistic regression models . . . Hosmer and Lemeshow have used very little mathematics, have presented difficult concepts heuristically and through illustrative examples, and have included references."—Choice"Well written, clearly organized, and comprehensive . . . the authors carefully walk the reader through the estimation of interpretation of coefficients from a wide variety of logistic regression models . . . their careful explication of the quantitative re-expression of coefficients from these various models is excellent."—Contemporary Sociology"An extremely well-written book that will certainly prove an invaluable acquisition to the practicing statistician who finds other literature on analysis of discrete data hard to follow or heavily theoretical."—The StatisticianIn this revised and updated edition of their popular book, David Hosmer and Stanley Lemeshow continue to provide an amazingly accessible introduction to the logistic regression model while incorporating advances of the last decade, including a variety of software packages for the analysis of data sets. Hosmer and Lemeshow extend the discussion from biostatistics and epidemiology to cutting-edge applications in data mining and machine learning, guiding readers step-by-step through the use of modeling techniques for dichotomous data in diverse fields. Ample new topics and expanded discussions of existing material are accompanied by a wealth of real-world examples-with extensive data sets available over the Internet.
A logistic regression model is proposed which enables one to model the boundary between growth and no growth for bacterial strains in the presence of one or more growth controlling factors such as temperature, pH and additives such as salt and sodium nitrite. The form of the expression containing the growth limiting factors may be suggested by a kinetic model, while the response at a given combination of factors may either be presence/absence (i.e. growth/no growth) or probabilistic (i.e. r successes in n trials). The approach described represents an integration of the probability and kinetic aspects of predictive microbiology, and a unification of predictive microbiology and the hurdle concept. The model is illustrated using data for Shigella flexneri.
A generalization of the coefficient of determination R2 to general regression models is discussed. A modification of an earlier definition to allow for discrete models is proposed.
A model was developed for the temperature dependence of growth rate of a non-pathogenic Escherichia coli strain. The suitability of that model for predicting the growth rate of pathogenic E. coli strains was assessed. Growth rates of pathogenic strains were found to be adequately described by the model. Model predictions were also found to describe sufficiently well-published growth rate data for non-pathogenic E. coli on mutton carcase surfaces and E. coli O157:H7 in ground roasted beef, milk, and on cantaloupes and water melons. In addition, E. coli O157:H7 was found to grow in the region of 44-45 degrees celsius.
A broth-based method is used to determine if exponential phase Escherichia coli R31, an STEC, is able to grow within 50 days under various combinations of sub-optimal temperatures and salt concentrations. From these data, the growth limits for combinations of temperature (7.7-37.0 degrees C) and water activity (0.943-0.987; NaCl as humectant) are defined and modelled using a nonlinear logistic regression model. That form of model is able to predict the combinations of salt concentration/water activity and temperature that will prevent the growth of E. coli R31 with selected levels of confidence. The model fitted the data with an approximate concordance rate of 97.3%. The minimum water activity that permitted growth occurred in the range 25-30 degrees C, the temperature range which optimises cell yield. At temperatures below this range the minimum water activity which allowed growth increased with decreasing temperature.
Some aspects of the ecology of Listeria monocytogenes in salmonid aquaculture
- S Tienungoon
Tienungoon, S., 1998. Some aspects of the ecology of Listeria monocytogenes in salmonid aquaculture. PhD thesis, University of Tasmania, Hobart, Tasmania, Australia.