International Journal of Food Microbiology 43 (1998) 97–104

Relation between the generation time and the lag time of bacterial growth kinetics M.L. Delignette-Muller* ´´ Laboratoire d’ Ecologie Microbienne et Parasitaire, Ecole Nationale Veterinaire de Lyon, 1 avenue Bourgelat, BP 83, 69280 Marcy ´ , France l’ etoile Received 13 February 1998; received in revised form 8 June 1998; accepted 15 June 1998

Abstract In predictive microbiology, the relation between the lag time (Lag) and the generation time (Tg) is commonly assumed to be proportional, as long as the pre-incubation environmental conditions remain constant. This relation was statistically examined in nine published datasets. For every dataset, it was roughly proportional. However, a more advanced study showed that the ratio Lag /Tg was not totally independent of the environmental conditions. In particular, a significant negative effect of the pH on this ratio was observed in five of the nine datasets. For modeling the environmental dependence of microbial growth parameters, some authors independently deal with Lag and Tg. Other authors only model the environmental dependence of Tg, assuming Lag /Tg to be constant. These two modeling methods were statistically compared for the nine datasets under study. Results differed from one dataset to another. For some, the model developed with a constant ratio Lag /Tg sufficed to describe the data, whereas for the others, an independent modeling of Lag and Tg was more satisfactory. 1998 Elsevier Science B.V. All rights reserved. Keywords: Predictive microbiology; Growth; Lag time; Generation time

1. Introduction Predictive microbiology is a subject of growing interest, which aims at forecasting the development of microorganisms in food using mathematical modeling. In predictive food microbiology, a bacterial growth curve is often characterized by two parameters: the lag time and the generation time. The lag time is the duration of the lag phase. It was *Corresponding author. Tel.: 133 4 78 87 25 92; fax: 133 4 78 87 25 94; e-mail: [email protected]

described decades ago by Buchanan (1918), and more recently it was mathematically defined by Buchanan and Cygnarowicz (1990) as the time when the second derivative of the logarithm of the growth curve reaches its maximum value. The generation time is the time needed for doubling the initial bacterial population during the exponential growth phase. It is directly linked to the specific growth rate, which is the slope of the logarithm of the growth curve in the exponential growth phase. These two parameters are generally estimated from the experimental growth kinetics by the fitting of a primary

0168-1605 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII: S0168-1605( 98 )00100-7

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M.L. Delignette-Muller / International Journal of Food Microbiology 43 (1998) 97 – 104

level model, such as the Gompertz modified equation (Gibson et al., 1988; Zwietering et al., 1990) or the model proposed by Baranyi and Roberts (1994). Secondary level models are then developed to describe the effect of some environmental conditions, such as temperature and pH, on these growth parameters. For this purpose, mainly two types of model are used, the polynomial-type models (Buchanan, 1991; Roberts, 1995) and the square root-type models (Ratkowsky et al., 1983; Zwietering et al., 1992; Rosso et al., 1995; McMeekin and Ross, 1996). Secondary level models do not always take the lag phase into consideration. Whenever they do so, it can be in different ways. Polynomial-type models are generally developed independently for the generation time and the lag time (Gibson et al., 1988; McClure et al., 1993; Buchanan and Bagi, 1994; Ng and Schaffner, 1997), whereas the square root-type models are generally based on the assumption that the lag time is proportional to the generation time (Adair et al., 1989; Ratkowsky et al., 1991; Zwietering et al., 1991). A proportional relation between the lag time and the generation time is implicitly or explicitly assumed by many microbiologists, but observations of this phenomenon on experimental data have rarely been published (Cooper, 1963; Baranyi and Roberts, 1994; McKellar, 1997). More precisely, it is generally admitted that, unlike the generation time, the lag time does not only depend on current growth conditions, but also on previous growth conditions (Pirt, 1975; Hudson, 1993; Dufrenne et al., 1997), but it is assumed that for cultures having identical physiological states at inoculation and being cultivated under different conditions, the ratio of the lag time to the generation time is constant. This idea is directly integrated into the primary level model proposed by Baranyi and Roberts (1994). In this model, the physiological state of the cells is involved as a new variable, and the ratio of the lag time to the generation time is presented as a simple function of the initial value of this variable. For the development of secondary level models, Baranyi and Roberts (1994) then proposed to use a simplified version of their primary level model, assuming the initial physiological state to be constant, and to model the environmental dependence of the generation time only.

As emphasized by Baranyi et al. (1996), parsimonious models should be preferred in predictive microbiology, due to their better robustness. Because a model with unnecessary parameters may become specific for the dataset to which it is fitted, the non-parsimonious models may have worse abilities in prediction than in adjustment (Delignette-Muller et al., 1995). Assuming a constant ratio of the lag to the generation time is then very appealing since it contributes to the minimization of the number of parameters. But, as mentioned by Baranyi and Roberts (1994), this simplification should be performed cautiously, and further research is necessary to decide in what region of environmental factors the lag and generation times are proportional. Within this context, the primary objective of this paper is to carefully examine the relation between the lag time and the generation time from published datasets that were used to develop secondary level models in predictive microbiology. This relation is statistically described, and its possible environmental dependence is studied. Different modeling strategies are then compared, in order to determine whether this relation should be taken into account or not.

2. Materials and methods Data were extracted from nine papers (Table 1). In these papers, the numerical data used to develop secondary level models for the lag time (Lag) and the generation time (Tg) was exhaustively given. In seven of the nine papers, the lag time and the generation time or the maximum specific growth rate [ mmax 5 ln(2) /Tg] were estimated by fitting the modified Gompertz equation (Gibson et al., 1988) to each experimental growth curve. The first version of the Baranyi equation (Baranyi et al., 1993) was used instead of the modified Gompertz equation in only one paper (McClure et al., 1993). Grau and Vanderlinde (1993) estimated the growth rate from linear regression analysis in the exponential phase of growth, and the lag time from the regression line as the time corresponding to the initial count of the inoculum. Microbial species, environmental factors under control and numbers of growth curves carried out in the experimental design are reported in Table 1. Many authors have discussed the choice of prob-

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Table 1 List of the analyzed datasets Dataset number

Microbial species

Reference

Control factors of growth

Total number of kinetics

Primary level model

1 2 3 4 5 6 7 8 9

Salmonella enterica Bacillus cereus Listeria monocytogenes Brochothrix thermosphacta Yersinia enterocolitica Escherichia coli O157:H7 Aeromonas hydrophila Shigella flexneri Bacillus stearothermophilus

Gibson et al., 1988 Benedict et al., 1993 Grau and Vanderlinde, 1993 McClure et al., 1993 Bhaduri et al., 1994 Buchanan and Bagi, 1994 McClure et al., 1994 ¨ et al., 1994 Zaıka Ng and Schaffner, 1997

T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH, NaCl, NaNO 2 T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH, NaCl

66 47 54 41 30 45 75 83 42

Gompertz Gompertz Log-linear with lag Baranyi Gompertz Gompertz Gompertz Gompertz Gompertz

ability distributions for modeling lag and generation time variability in statistical analyses (Ratkowsky et al., 1991; Alber and Schaffner, 1992; Schaffner, 1994; Zwietering et al., 1994; Ratkowsky et al., 1996; Ng and Schaffner, 1997). As no unanimous result has issued from these studies, we simply chose the logarithmic transformation to stabilize both lag and generation time variances. This simple transformation has commonly been used for the fitting of secondary level polynomial type models, and has been recommended by several authors for modeling generation time variability (Alber and Schaffner, 1992; Ratkowsky et al., 1996) and lag time variability (Ratkowsky et al., 1991; Alber and Schaffner, 1992; Zwietering et al., 1994). For each dataset, the relationship between ln(Lag) and ln(Tg) was studied by linear regression analysis using the StatView v.4.5 software (Abacus Concepts, Berkeley, CA, USA). In order to know whether environmental factors could explain a part of the variability of the ratio Lag /Tg, a quadratic polynomial regression was performed on ln(Lag /Tg). A forward stepwise procedure was used to select significant factors with the StatView software. A global secondary level model for Lag and Tg can be written as: y 5 f(a,c) with

H

y 5 lnsLagd if c 5 1 y 5 lnsTgd if c 5 0

(1)

where a is the vector of the environmental factors (a5(T, pH,...)) taken into account. For developing polynomial type models, the function f is commonly defined (as below), as the sum of two polynomial

functions of the environmental factors independently describing Lag and Tg. y 5s1 2 cd 3 fTgsad 1 c 3 fLagsad or y 5 fTgsad 1 c 3 fLag 2Tgsad

(2)

If the ratio Lag /Tg is assumed to be constant, the function f can be more simply defined as below, with b Lag 2 Tg corresponding to the constant value of ln(Lag /Tg). y 5 fTgsad 1 c 3 b Lag 2Tg

(3)

Model (3) is nested into Model (2). It is obtained by fixing the function fLag 2 Tg to the constant b Lag 2 Tg . The F test comparing two nested models (Bates and Watts, 1988; Zwietering et al., 1990) can be used to check whether the simplified Model (3) would suffice to describe the data. For each dataset, two quadratic polynomial models corresponding to Eqs. (2) and (3) were developed and statistically compared using this F test.

3. Results For each dataset (Table 1), bivariate scattergrams are plotted to display the relationship between ln(Lag) and ln(Tg) (Fig. 1). The regression lines obtained with ln(Lag) as the dependent variable and ln(Tg) as the independent one are added to these scattergrams, as well as the regression equation and the coefficient of determination (r 2 value). For each dataset, a significant linear correlation is observed between ln(Lag) and ln(Tg). For a strictly constant ratio of Lag /Tg, the slope of the regression lines

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Fig. 1. Relation between ln(Lag) and ln(Tg) for the nine datasets.

should theoretically be equal to one. The observed values of this slope are close to one (between 0.91 and 1.14) and their confidence intervals always contain the theoretical value of one. This reinforces the hypothesis of a constant ratio of Lag /Tg. The box plots displaying the 10th, 25th, 50th, 75th and 90th percentiles of ln(Lag /Tg) in each dataset are plotted in Fig. 2. Average values from 0.22 to 2.72 were observed for ln(Lag /Tg).

Environmental factors selected by stepwise regression to explain the dependent variable ln(Lag /Tg) are reported in Table 2. We also report adjusted r 2 values, a widely used criterion characterizing the percentage variance accounted for. The pH was selected in six of the nine datasets. Five times within these six datasets, there is a clear negative correlation between ln(Lag /Tg) and pH, three times with a strong statistical significance ( p,0.001). The so-

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Fig. 2. Distribution of ln(Lag /Tg) for the nine datasets (box plots displaying the 10th, 25th, 50th, 75th and 90th percentiles of the variable).

dium chloride concentration was selected in four of the eight datasets involving this factor, twice with a clear positive correlation between ln(Lag /Tg) and NaCl (%). The temperature was selected in only two of the nine datasets, and never with a simple correlation between ln(Lag /Tg) and temperature. The

sodium nitrite concentration was never selected in the four datasets involving this factor. In only one of the nine datasets does the adjusted r 2 value exceed 50%. This shows that a major part of the variability of ln(Lag /Tg) cannot be explained by the classical environmental factors.

Table 2 Environmental factors selected to explain the variability of ln(Lag /Tg) Dataset number

Selected regression equation

Adjusted r 2

1 2 3 4 5 6 7 8 9

1.0690410.0214 NaCl 2 a No selected factors 6.5697–1.0669 pH c 3.849010.1036 NaCl b –0.4719 pH 2 a No selected factors 5.4404–0.5782 pH c 10.0007 T3NaCl c 6.2153–0.3826 pH c –0.3137 T c 10.0068 T 2 c 10.1504 NaCl 2 c –0.4285 NaCl b 10.0181 pH3T a 3.8084–0.3204 pH a 3.1956–0.2711 pH c

5.6%

The level of statistical significance of each selected regression factor is indicated by: a p,0.05, b p,0.01 and c p,0.001.

46.9% 25.2% 84.2% 48.5% 4.0% 22.8%

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Table 3 Results of the F test comparing Models (2) and (3) Dataset

Adjusted r 2 with Model (2)

Adjusted r 2 with Model (3)

f value

1 2 3 4 5 6 7 8 9

95.5% 80.4% 95.3% 90.6% 91.2% 89.5% 97.2% 69.7% 94.9%

95.1% 82.0% 91.5% 88.6% 90.0% 86.9% 96.2% 70.8% 94.3%

2.32 0.54 16.92 2.69 1.44 2.35 6.11 0.59 1.83

a

c a

a c

F (table)

1.96 1.85 2.31 2.03 2.04 1.86 1.95 1.77 2.03

The level of statistical significance of each observed f value is indicated by: a p,0.05, b p,0.01 and c p,0.001.

The results of the comparison of Models (2) and (3), reported in Table 3, indicate that it is worthwhile using the less parsimonious Model (2) to reduce the RSS (residual sums of squares) in five datasets among nine. For the four other datasets, the simplified Model (3), which assumes a constant ratio of Lag /Tg, suffices to describe the Lag and Tg data. In all cases, the goodness of fit of the simplified model remains satisfactory. The difference between the two adjusted r 2 values never exceeds 4%.

4. Discussion This analysis of nine experimental datasets reinforces the generally accepted idea that the lag time is proportional to the generation time as long as the pre-incubation environmental conditions remain constant. The statistical distribution of the ratio Lag /Tg is generally closer to a lognormal distribution than to a normal distribution (Fig. 2). As a consequence, in a method such as the one proposed by Baranyi and Roberts (1994), its constant value should merely be estimated from the average value of ln(Lag /Tg). The variability of the average values of ln(Lag /Tg) observed on Fig. 2 may be at least partially explained by the different pre-incubation conditions corresponding to each dataset. In particular, it is worth noting that the small value observed for the third dataset corresponds to the only study where the pre-incubation temperature was far from the optimal temperature. The Listeria monocytogenes strain (Grau and Vanderlinde, 1993) was pre-incubated at

108C, whereas the strains studied in the other papers were pre-incubated at temperatures higher than 208C. This observation matches with a recent study of the effect of the pre-incubation temperature on the lag phase duration of some foodborne pathogenic microorganisms (Dufrenne et al., 1997), where low values of the ratio Lag /Tg were observed for low preincubation temperatures. This comment is no more than a possible explanation, since the dataset of Grau and Vanderlinde (1993) also differs from the other datasets in the culture medium used, being made of lean and fatty beef tissue. Moreover, the physiological state (lag, exponential, stationary) of the inoculum differs from one dataset to another, which could also explain a part of the variability of the average values of ln(Lag /Tg). For some datasets, the ratio Lag /Tg is not totally independent of post-incubation environmental conditions. In particular, a negative linear correlation between the variable ln(Lag /Tg) and the pH is observed in more than half of the nine datasets. This observation may be related to a physiological stress of the cells induced by their introduction into an acid medium. This stress could make the lag phase longer than expected, so increasing the ratio. Assuming that the ratio Lag /Tg is constant then appears as an approximation. One of the questions raised by this study concerns a better way to model the effects of environmental factors on Lag and Tg. Is it better to assume that these parameters are proportional or to model them independently? The results of our study show that, in some cases, the first alternative seems the best, whereas in other cases, the second seems the best. What then is the right solution? Deciding not to focus on the relation linking Lag and Tg will often lead to models with too many unnecessary parameters. Even after a stepwise selection procedure, Model (2) generally keeps more parameters than Model (3), at least for the nine datasets studied in this paper. Therefore, this solution is not optimal if we aim at developing parsimonious models, which are more effective in prediction (Baranyi et al., 1996; Delignette-Muller et al., 1995). Deliberately assuming a constant ratio of Lag /Tg in all cases can then be considered as a solution. This oversimplification gives more parsimonious models, without sacrificing too much of their goodness of fit. Moreover, this simplification has an important conse-

M.L. Delignette-Muller / International Journal of Food Microbiology 43 (1998) 97 – 104

quence for use in prediction. Estimating the constant ratio Lag /Tg suffices for determining the lag time from the generation time. As the generation time only depends on the post-incubation environmental conditions, the correct use of a secondary level model in different pre-incubation conditions only requires the estimation of the new ratio Lag /Tg under these new conditions. Nevertheless, this simplification leads to the neglecting of some possible environmental effects on the relation between the lag time and the generation time, such as the previously observed effect of the pH. Modeling methods requiring this simplification, such as the one proposed by Baranyi and Roberts (1994), should then be used cautiously. Before simplifying the model by fixing the ratio Lag /Tg to a constant value, one should check if there is no clear dependence of this ratio on the environmental factors. If the case occurs, one should quantify the impact of the simplification on the goodness of fit of the model. These precautions being taken, such a method offers a simple way to reduce the number of the parameters of the model. It also makes the use of models in prediction easier whenever possible.

Acknowledgements ´ ´ and S. Breand ´ I wish to thank V. Guerin-Faublee for critically reading the manuscript and making several pertinent remarks.

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Ng, T.M., Schaffner, D.W., 1997. Mathematical models for the effects of pH, temperature, and sodium chloride on the growth of Bacillus stearothermophilus in salty carrots. Appl. Environ. Microbiol. 63, 1237–1243. Pirt, S.J., 1975. Growth lag, In: Principles of Microbe and Cell Cultivation, Blackwell, Oxford, pp. 194–198. Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N., Chandler, R.E., 1983. Model for bacterial culture growth rate throughout the entire biokinetic temperature range. J. Bacteriol. 154, 1222–1226. Ratkowsky, D.A., Ross, T., McMeekin, T.A., Olley, J., 1991. Comparison of Arrhenius-type and Belehradek-type models for prediction of bacterial growth in foods. J. Appl. Bacteriol. 71, 452–459. Ratkowsky, D.A., Ross, T., Macario, N., Dommett, T.W., Kamperman, L., 1996. Choosing probability distributions for modelling generation time variability. J. Appl. Bacteriol. 80, 131– 137. Roberts, T.A., 1995. Microbial growth and survival: developments in predictive modelling, Int. Biodeter. Biodegr. 297–309. Rosso, L., Lobry, J.R., Bajard, S., Flandrois, J.P., 1995. Convenient model to describe the combined effects of temperature and pH on microbial growth. Appl. Environ. Microbiol. 61, 610– 616.

Schaffner, D.W., 1994. Application of a statistical bootstrapping technique to calculate growth rate variance for modelling psychrotrophic pathogen growth. Int. J. Food Microbiol. 24, 309–314. ¨ Zaıka, L.L., Moulden, E., Weimer, L., Phillips, J.G., Buchanan, R.L., 1994. Model for the combined effects of temperature, initial pH, sodium chloride and sodium nitrite concentrations on anaerobic growth of Shigella flexneri. Int. J. Food Microbiol. 23, 345–358. Zwietering, M.H., Jongenburger, I., Rombouts, F.M., van’t Riet, K., 1990. Modeling of the bacterial growth curve. Appl. Environ. Microbiol. 56, 1875–1881. Zwietering, M.H., de Koos, J.T., Hasenack, B.E., de Wit, J.C., van’t Riet, K., 1991. Modeling of bacterial growth as a function of temperature. Appl. Environ. Microbiol. 57, 1094– 1101. Zwietering, M.H., Wijtzes, T., de Wit, J.C., van’t Riet, K., 1992. A decision support system for prediction of the microbial spoilage in foods. J. Food Protect. 55, 973–979. Zwietering, M.H., Cuppers, H.G.A.M., de Wit, J.C., van’t Riet, K., 1994. Evaluation of data transformations and validation of a model for the effect of temperature on bacterial growth. Appl. Environ. Microbiol. 60, 195–203.

Relation between the generation time and the lag time of bacterial growth kinetics M.L. Delignette-Muller* ´´ Laboratoire d’ Ecologie Microbienne et Parasitaire, Ecole Nationale Veterinaire de Lyon, 1 avenue Bourgelat, BP 83, 69280 Marcy ´ , France l’ etoile Received 13 February 1998; received in revised form 8 June 1998; accepted 15 June 1998

Abstract In predictive microbiology, the relation between the lag time (Lag) and the generation time (Tg) is commonly assumed to be proportional, as long as the pre-incubation environmental conditions remain constant. This relation was statistically examined in nine published datasets. For every dataset, it was roughly proportional. However, a more advanced study showed that the ratio Lag /Tg was not totally independent of the environmental conditions. In particular, a significant negative effect of the pH on this ratio was observed in five of the nine datasets. For modeling the environmental dependence of microbial growth parameters, some authors independently deal with Lag and Tg. Other authors only model the environmental dependence of Tg, assuming Lag /Tg to be constant. These two modeling methods were statistically compared for the nine datasets under study. Results differed from one dataset to another. For some, the model developed with a constant ratio Lag /Tg sufficed to describe the data, whereas for the others, an independent modeling of Lag and Tg was more satisfactory. 1998 Elsevier Science B.V. All rights reserved. Keywords: Predictive microbiology; Growth; Lag time; Generation time

1. Introduction Predictive microbiology is a subject of growing interest, which aims at forecasting the development of microorganisms in food using mathematical modeling. In predictive food microbiology, a bacterial growth curve is often characterized by two parameters: the lag time and the generation time. The lag time is the duration of the lag phase. It was *Corresponding author. Tel.: 133 4 78 87 25 92; fax: 133 4 78 87 25 94; e-mail: [email protected]

described decades ago by Buchanan (1918), and more recently it was mathematically defined by Buchanan and Cygnarowicz (1990) as the time when the second derivative of the logarithm of the growth curve reaches its maximum value. The generation time is the time needed for doubling the initial bacterial population during the exponential growth phase. It is directly linked to the specific growth rate, which is the slope of the logarithm of the growth curve in the exponential growth phase. These two parameters are generally estimated from the experimental growth kinetics by the fitting of a primary

0168-1605 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII: S0168-1605( 98 )00100-7

98

M.L. Delignette-Muller / International Journal of Food Microbiology 43 (1998) 97 – 104

level model, such as the Gompertz modified equation (Gibson et al., 1988; Zwietering et al., 1990) or the model proposed by Baranyi and Roberts (1994). Secondary level models are then developed to describe the effect of some environmental conditions, such as temperature and pH, on these growth parameters. For this purpose, mainly two types of model are used, the polynomial-type models (Buchanan, 1991; Roberts, 1995) and the square root-type models (Ratkowsky et al., 1983; Zwietering et al., 1992; Rosso et al., 1995; McMeekin and Ross, 1996). Secondary level models do not always take the lag phase into consideration. Whenever they do so, it can be in different ways. Polynomial-type models are generally developed independently for the generation time and the lag time (Gibson et al., 1988; McClure et al., 1993; Buchanan and Bagi, 1994; Ng and Schaffner, 1997), whereas the square root-type models are generally based on the assumption that the lag time is proportional to the generation time (Adair et al., 1989; Ratkowsky et al., 1991; Zwietering et al., 1991). A proportional relation between the lag time and the generation time is implicitly or explicitly assumed by many microbiologists, but observations of this phenomenon on experimental data have rarely been published (Cooper, 1963; Baranyi and Roberts, 1994; McKellar, 1997). More precisely, it is generally admitted that, unlike the generation time, the lag time does not only depend on current growth conditions, but also on previous growth conditions (Pirt, 1975; Hudson, 1993; Dufrenne et al., 1997), but it is assumed that for cultures having identical physiological states at inoculation and being cultivated under different conditions, the ratio of the lag time to the generation time is constant. This idea is directly integrated into the primary level model proposed by Baranyi and Roberts (1994). In this model, the physiological state of the cells is involved as a new variable, and the ratio of the lag time to the generation time is presented as a simple function of the initial value of this variable. For the development of secondary level models, Baranyi and Roberts (1994) then proposed to use a simplified version of their primary level model, assuming the initial physiological state to be constant, and to model the environmental dependence of the generation time only.

As emphasized by Baranyi et al. (1996), parsimonious models should be preferred in predictive microbiology, due to their better robustness. Because a model with unnecessary parameters may become specific for the dataset to which it is fitted, the non-parsimonious models may have worse abilities in prediction than in adjustment (Delignette-Muller et al., 1995). Assuming a constant ratio of the lag to the generation time is then very appealing since it contributes to the minimization of the number of parameters. But, as mentioned by Baranyi and Roberts (1994), this simplification should be performed cautiously, and further research is necessary to decide in what region of environmental factors the lag and generation times are proportional. Within this context, the primary objective of this paper is to carefully examine the relation between the lag time and the generation time from published datasets that were used to develop secondary level models in predictive microbiology. This relation is statistically described, and its possible environmental dependence is studied. Different modeling strategies are then compared, in order to determine whether this relation should be taken into account or not.

2. Materials and methods Data were extracted from nine papers (Table 1). In these papers, the numerical data used to develop secondary level models for the lag time (Lag) and the generation time (Tg) was exhaustively given. In seven of the nine papers, the lag time and the generation time or the maximum specific growth rate [ mmax 5 ln(2) /Tg] were estimated by fitting the modified Gompertz equation (Gibson et al., 1988) to each experimental growth curve. The first version of the Baranyi equation (Baranyi et al., 1993) was used instead of the modified Gompertz equation in only one paper (McClure et al., 1993). Grau and Vanderlinde (1993) estimated the growth rate from linear regression analysis in the exponential phase of growth, and the lag time from the regression line as the time corresponding to the initial count of the inoculum. Microbial species, environmental factors under control and numbers of growth curves carried out in the experimental design are reported in Table 1. Many authors have discussed the choice of prob-

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Table 1 List of the analyzed datasets Dataset number

Microbial species

Reference

Control factors of growth

Total number of kinetics

Primary level model

1 2 3 4 5 6 7 8 9

Salmonella enterica Bacillus cereus Listeria monocytogenes Brochothrix thermosphacta Yersinia enterocolitica Escherichia coli O157:H7 Aeromonas hydrophila Shigella flexneri Bacillus stearothermophilus

Gibson et al., 1988 Benedict et al., 1993 Grau and Vanderlinde, 1993 McClure et al., 1993 Bhaduri et al., 1994 Buchanan and Bagi, 1994 McClure et al., 1994 ¨ et al., 1994 Zaıka Ng and Schaffner, 1997

T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH, NaCl, NaNO 2 T, pH, NaCl T, pH, NaCl, NaNO 2 T, pH, NaCl

66 47 54 41 30 45 75 83 42

Gompertz Gompertz Log-linear with lag Baranyi Gompertz Gompertz Gompertz Gompertz Gompertz

ability distributions for modeling lag and generation time variability in statistical analyses (Ratkowsky et al., 1991; Alber and Schaffner, 1992; Schaffner, 1994; Zwietering et al., 1994; Ratkowsky et al., 1996; Ng and Schaffner, 1997). As no unanimous result has issued from these studies, we simply chose the logarithmic transformation to stabilize both lag and generation time variances. This simple transformation has commonly been used for the fitting of secondary level polynomial type models, and has been recommended by several authors for modeling generation time variability (Alber and Schaffner, 1992; Ratkowsky et al., 1996) and lag time variability (Ratkowsky et al., 1991; Alber and Schaffner, 1992; Zwietering et al., 1994). For each dataset, the relationship between ln(Lag) and ln(Tg) was studied by linear regression analysis using the StatView v.4.5 software (Abacus Concepts, Berkeley, CA, USA). In order to know whether environmental factors could explain a part of the variability of the ratio Lag /Tg, a quadratic polynomial regression was performed on ln(Lag /Tg). A forward stepwise procedure was used to select significant factors with the StatView software. A global secondary level model for Lag and Tg can be written as: y 5 f(a,c) with

H

y 5 lnsLagd if c 5 1 y 5 lnsTgd if c 5 0

(1)

where a is the vector of the environmental factors (a5(T, pH,...)) taken into account. For developing polynomial type models, the function f is commonly defined (as below), as the sum of two polynomial

functions of the environmental factors independently describing Lag and Tg. y 5s1 2 cd 3 fTgsad 1 c 3 fLagsad or y 5 fTgsad 1 c 3 fLag 2Tgsad

(2)

If the ratio Lag /Tg is assumed to be constant, the function f can be more simply defined as below, with b Lag 2 Tg corresponding to the constant value of ln(Lag /Tg). y 5 fTgsad 1 c 3 b Lag 2Tg

(3)

Model (3) is nested into Model (2). It is obtained by fixing the function fLag 2 Tg to the constant b Lag 2 Tg . The F test comparing two nested models (Bates and Watts, 1988; Zwietering et al., 1990) can be used to check whether the simplified Model (3) would suffice to describe the data. For each dataset, two quadratic polynomial models corresponding to Eqs. (2) and (3) were developed and statistically compared using this F test.

3. Results For each dataset (Table 1), bivariate scattergrams are plotted to display the relationship between ln(Lag) and ln(Tg) (Fig. 1). The regression lines obtained with ln(Lag) as the dependent variable and ln(Tg) as the independent one are added to these scattergrams, as well as the regression equation and the coefficient of determination (r 2 value). For each dataset, a significant linear correlation is observed between ln(Lag) and ln(Tg). For a strictly constant ratio of Lag /Tg, the slope of the regression lines

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Fig. 1. Relation between ln(Lag) and ln(Tg) for the nine datasets.

should theoretically be equal to one. The observed values of this slope are close to one (between 0.91 and 1.14) and their confidence intervals always contain the theoretical value of one. This reinforces the hypothesis of a constant ratio of Lag /Tg. The box plots displaying the 10th, 25th, 50th, 75th and 90th percentiles of ln(Lag /Tg) in each dataset are plotted in Fig. 2. Average values from 0.22 to 2.72 were observed for ln(Lag /Tg).

Environmental factors selected by stepwise regression to explain the dependent variable ln(Lag /Tg) are reported in Table 2. We also report adjusted r 2 values, a widely used criterion characterizing the percentage variance accounted for. The pH was selected in six of the nine datasets. Five times within these six datasets, there is a clear negative correlation between ln(Lag /Tg) and pH, three times with a strong statistical significance ( p,0.001). The so-

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Fig. 2. Distribution of ln(Lag /Tg) for the nine datasets (box plots displaying the 10th, 25th, 50th, 75th and 90th percentiles of the variable).

dium chloride concentration was selected in four of the eight datasets involving this factor, twice with a clear positive correlation between ln(Lag /Tg) and NaCl (%). The temperature was selected in only two of the nine datasets, and never with a simple correlation between ln(Lag /Tg) and temperature. The

sodium nitrite concentration was never selected in the four datasets involving this factor. In only one of the nine datasets does the adjusted r 2 value exceed 50%. This shows that a major part of the variability of ln(Lag /Tg) cannot be explained by the classical environmental factors.

Table 2 Environmental factors selected to explain the variability of ln(Lag /Tg) Dataset number

Selected regression equation

Adjusted r 2

1 2 3 4 5 6 7 8 9

1.0690410.0214 NaCl 2 a No selected factors 6.5697–1.0669 pH c 3.849010.1036 NaCl b –0.4719 pH 2 a No selected factors 5.4404–0.5782 pH c 10.0007 T3NaCl c 6.2153–0.3826 pH c –0.3137 T c 10.0068 T 2 c 10.1504 NaCl 2 c –0.4285 NaCl b 10.0181 pH3T a 3.8084–0.3204 pH a 3.1956–0.2711 pH c

5.6%

The level of statistical significance of each selected regression factor is indicated by: a p,0.05, b p,0.01 and c p,0.001.

46.9% 25.2% 84.2% 48.5% 4.0% 22.8%

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Table 3 Results of the F test comparing Models (2) and (3) Dataset

Adjusted r 2 with Model (2)

Adjusted r 2 with Model (3)

f value

1 2 3 4 5 6 7 8 9

95.5% 80.4% 95.3% 90.6% 91.2% 89.5% 97.2% 69.7% 94.9%

95.1% 82.0% 91.5% 88.6% 90.0% 86.9% 96.2% 70.8% 94.3%

2.32 0.54 16.92 2.69 1.44 2.35 6.11 0.59 1.83

a

c a

a c

F (table)

1.96 1.85 2.31 2.03 2.04 1.86 1.95 1.77 2.03

The level of statistical significance of each observed f value is indicated by: a p,0.05, b p,0.01 and c p,0.001.

The results of the comparison of Models (2) and (3), reported in Table 3, indicate that it is worthwhile using the less parsimonious Model (2) to reduce the RSS (residual sums of squares) in five datasets among nine. For the four other datasets, the simplified Model (3), which assumes a constant ratio of Lag /Tg, suffices to describe the Lag and Tg data. In all cases, the goodness of fit of the simplified model remains satisfactory. The difference between the two adjusted r 2 values never exceeds 4%.

4. Discussion This analysis of nine experimental datasets reinforces the generally accepted idea that the lag time is proportional to the generation time as long as the pre-incubation environmental conditions remain constant. The statistical distribution of the ratio Lag /Tg is generally closer to a lognormal distribution than to a normal distribution (Fig. 2). As a consequence, in a method such as the one proposed by Baranyi and Roberts (1994), its constant value should merely be estimated from the average value of ln(Lag /Tg). The variability of the average values of ln(Lag /Tg) observed on Fig. 2 may be at least partially explained by the different pre-incubation conditions corresponding to each dataset. In particular, it is worth noting that the small value observed for the third dataset corresponds to the only study where the pre-incubation temperature was far from the optimal temperature. The Listeria monocytogenes strain (Grau and Vanderlinde, 1993) was pre-incubated at

108C, whereas the strains studied in the other papers were pre-incubated at temperatures higher than 208C. This observation matches with a recent study of the effect of the pre-incubation temperature on the lag phase duration of some foodborne pathogenic microorganisms (Dufrenne et al., 1997), where low values of the ratio Lag /Tg were observed for low preincubation temperatures. This comment is no more than a possible explanation, since the dataset of Grau and Vanderlinde (1993) also differs from the other datasets in the culture medium used, being made of lean and fatty beef tissue. Moreover, the physiological state (lag, exponential, stationary) of the inoculum differs from one dataset to another, which could also explain a part of the variability of the average values of ln(Lag /Tg). For some datasets, the ratio Lag /Tg is not totally independent of post-incubation environmental conditions. In particular, a negative linear correlation between the variable ln(Lag /Tg) and the pH is observed in more than half of the nine datasets. This observation may be related to a physiological stress of the cells induced by their introduction into an acid medium. This stress could make the lag phase longer than expected, so increasing the ratio. Assuming that the ratio Lag /Tg is constant then appears as an approximation. One of the questions raised by this study concerns a better way to model the effects of environmental factors on Lag and Tg. Is it better to assume that these parameters are proportional or to model them independently? The results of our study show that, in some cases, the first alternative seems the best, whereas in other cases, the second seems the best. What then is the right solution? Deciding not to focus on the relation linking Lag and Tg will often lead to models with too many unnecessary parameters. Even after a stepwise selection procedure, Model (2) generally keeps more parameters than Model (3), at least for the nine datasets studied in this paper. Therefore, this solution is not optimal if we aim at developing parsimonious models, which are more effective in prediction (Baranyi et al., 1996; Delignette-Muller et al., 1995). Deliberately assuming a constant ratio of Lag /Tg in all cases can then be considered as a solution. This oversimplification gives more parsimonious models, without sacrificing too much of their goodness of fit. Moreover, this simplification has an important conse-

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quence for use in prediction. Estimating the constant ratio Lag /Tg suffices for determining the lag time from the generation time. As the generation time only depends on the post-incubation environmental conditions, the correct use of a secondary level model in different pre-incubation conditions only requires the estimation of the new ratio Lag /Tg under these new conditions. Nevertheless, this simplification leads to the neglecting of some possible environmental effects on the relation between the lag time and the generation time, such as the previously observed effect of the pH. Modeling methods requiring this simplification, such as the one proposed by Baranyi and Roberts (1994), should then be used cautiously. Before simplifying the model by fixing the ratio Lag /Tg to a constant value, one should check if there is no clear dependence of this ratio on the environmental factors. If the case occurs, one should quantify the impact of the simplification on the goodness of fit of the model. These precautions being taken, such a method offers a simple way to reduce the number of the parameters of the model. It also makes the use of models in prediction easier whenever possible.

Acknowledgements ´ ´ and S. Breand ´ I wish to thank V. Guerin-Faublee for critically reading the manuscript and making several pertinent remarks.

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