no-growth interface and nonthermal inactivation areas of Listeria in foods

International Journal of Food Microbiology 152 (2012) 139–152

Contents lists available at SciVerse ScienceDirect

International Journal of Food Microbiology journal homepage: www.elsevier.com/locate/ijfoodmicro

Modelling of growth, growth/no-growth interface and nonthermal inactivation areas of Listeria in foods Louis Coroller a,⁎, Denis Kan-King-Yu b, Ivan Leguerinel a, Pierre Mafart a, Jeanne-Marie Membré c, d a Université Européenne de Bretagne, France—Université de Brest, EA3882, Laboratoire Universitaire de Biodiversité et Ecologie Microbienne, IFR148 ScInBioS, UMT 08.3 PHYSI'Opt, 6 rue de l'Université, F-29334 Quimper, France b Unilever R&D Colworth—Safety & Environmental Assurance Centre, Colworth Park, Sharnbrook MK44 1LQ, UK c INRA, UMR 1014 SECALIM, Rue de la Géraudière, BP 82225, 44332 NANTES Cedex 3, France d LUNAM Université, Oniris, Nantes, F-44307, France

a r t i c l e

i n f o

Available online 2 October 2011 Keywords: Predictive microbiology Growth rate Growth boundary Survival Inactivation

a b s t r a c t Growth, growth boundary and inactivation models have been extensively developed in predictive microbiology and are commonly applied in food research nowadays. Few studies though report the development of models which encompass all three areas together. A tiered modelling approach, based on the Gamma hypothesis, is proposed here to predict the behaviour of Listeria. Datasets of Listeria spp. behaviour in laboratory media, meat, dairy, seafood products and vegetables were collected from literature, unpublished sources and from the databases ComBase and Sym'Previus. The explanatory factors were temperature, pH, water activity, lactic and sorbic acids. For the growth part, 697 growth kinetic datasets were fitted. The estimated growth rates and 2021 additional growth primary datasets were used to fit the secondary growth models. In a second step, the fitted model was used to predict the growth/no-growth boundary. For the inactivation modelling phase, 535 inactivation curves were used. Gamma models with and without interactions between the explanatory factors were used for the growth and boundary models. The correct prediction percentage (predicted growth when growth is observed + predicted inactivation when inactivation is observed) varied from 62% to 81% for the models without interactions, and from 85% to 87% for the models with interactions. The median error for the predicted population size was less than 0.34 log10(CFU/mL) for all models. The kinetics of inactivation were fitted with modified Weibull primary models and the estimated bacterial resistance was then modelled as a function of the explanatory factors. The error for the predicted microbial population size was less than 0.71 log10(CFU/mL) with a median value of less than 0.21 for all foods. The model enables the quantification of the increase or decrease in the bacterial population for a given formulation or storage condition. It might also be used to optimise a food formulation or storage condition in the case of a targeted increase or decrease of the bacterial population. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The vast majority of the models reported in Predictive Microbiology have been historically developed with a view to describing either microbial growth or inactivation. In the context of food safety, microbiologists are primarily interested in ensuring that pathogenic microorganisms are killed or restricted to a required level in their products (at a specific time point). Moreover, if the product is safe by design, it becomes natural to concentrate on spoilage microorganisms to make

⁎ Corresponding author at: Université Européenne de Bretagne, France—Université de Brest, EA3882, Laboratoire Universitaire de Biodiversité et Ecologie Microbienne, IFR148 ScInBioS, UMT 08.3 PHYSI'Opt. 6, rue de l'université, 29334 Quimper, France. Tel.: + 33 2 98 64 19 30; fax: + 33 2 98 64 19 69. E-mail address: [email protected] (L. Coroller). 0168-1605/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2011.09.023

sure that they are not able to grow to the extent of spoiling the foodstuff before the end of shelf-life (food quality/stability). The recent development of the Codex Alimentarius Guidelines on the Application of General principles of Food Hygiene to the Control of Listeria monocytogenes in Ready-To-Eat Foods (CAC/GL 61–2007), formally adopted in June 2009, has highlighted the importance of controlling this important pathogen in foods. The purpose of these guidelines is to provide guidance on the controls and associated tools that can be adopted by regulators and industry to minimise the risk of listeriosis from ready-to-eat foods. In addition, there have also been increases in cases of listeriosis in the elderly in Belgium, Denmark, England, Wales and Finland (Goulet et al., 2008). In the UK, these cases have been linked to ready-to-eat foods with an extended (usually refrigerated) shelf-life, able to support growth of L. monocytogenes (ACMSF, 2009). It is the responsibility of food

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manufacturers to demonstrate the growth/no-growth potential for L. monocytogenes in their ready-to-eat products and it is absolutely critical that effective controls are in place. With consumer demands for fresher, more natural and nutritious foods, the food industry is faced with the challenge of providing minimally-processed foods with less preservatives, that are still safe to eat. This challenge involves formulating foods at conditions that are becoming closer to the microbial growth/no-growth boundaries, where either microbial inactivation or (slow) growth could occur. Therefore, it becomes critical to understand and quantify the interaction effects of different hurdle technologies (Leistner, 2000) on microbial population behaviour. Few combined models describing changes in a microbial population subjected to conditions that vary from growth to inactivation have been reported (Corradini and Peleg, 2006; Jones et al., 1994; Ross et al., 2005; Whiting and Cygnarowicz-Provost, 1992). This article presents a tiered modelling approach—based on the Gamma hypothesis—that predicts the global behaviour of Listeria in various media. The proposed model postulates that only two microbial responses can be observed: growth or inactivation. When the maximum growth rate (as estimated from the Gamma concept) is greater than zero, microbial growth is predicted. When the maximum growth rate is equal to zero, then the bacteri-

al population is inactivated. This postulate is based on the following observation. In the literature, microbial responses are typically classified as growth, survival and inactivation. These three responses are time dependent and it can be reasonably extrapolated that if the bacterial behaviour was observed in static conditions for an infinite time period, only growth or inactivation would be observed. Microbial survival would therefore be characterised by either slow growth or slow inactivation and the concept of infinite lag would have no meaning in this context. L. monocytogenes was selected on the basis that it remains a serious threat for the safety of ready-to-eat foods and numerous data have been published and are available. The environmental factors of interest were temperature, pH, sorbic acid, lactic acid and sodium chloride salt. For further application in a industrial research and development context, the modelling approach investigated in this study had to consider the three following constraints. The biological parameters that are used in the model should be easily found in publications or elicited by an expert. The model should allow the prediction of bacterial behaviour (i.e. log increase or decrease) from the food formulation and storage condition and vice-versa, i.e. it should provide the required food properties given the targeted bacterial behaviour, or it should provide the storage condition from the food formulation and the targeted bacterial behaviour.

2. Materials and methods

2.1. The database The data used in this study were collected from publications, unpublished works of research institutes and Unilever (Table 1). 2319 kinetics and 2125 sets of fitted primary parameters were collected. The collected data described growth, survival or inactivation of Listeria spp. in 132 media: 50 types of microbiological broth, 30 meat products, 20 milk products, 11 seafood products, 13 vegetable or fruit products, 4 sauces or salad dressings, 2 egg products and 2 cake products. These data were associated with experimental factors including temperature, pH, sodium chloride concentration or water activity, sorbic and lactic acid concentration. When no information was provided, the acid concentration was assumed to be zero, and the water activity equal to 0.996. The water activity (aw) was recalculated from the sodium chloride concentration using the equation of Resnik and Chirife (1988) and vice-versa (Augustin et al., 2005; Giménez and Dalgaard, 2004; Theys et al., 2010):

aw ¼ 1−0:0052471:WPS−0:00012206:WSP

With WPS ¼

100:%NaCl %moisture þ %NaCl

2

ð1Þ

ð2Þ

Where WPS is the water phase salt calculated from the sodium chloride concentration in percentage (%NaCl, W/V) and the water in the product (%moisture, W/V). At this stage, the data were selected based on storage temperature values from 0 °C to 35 °C, pH from 3.0 to 6.5, a sodium chloride concentration of less than 15% (w/w) or water activity over 0.90, sorbic acid less than 0.1% total acid and lactic acid less than 2% total acid. Kinetic data over six points were kept for primary modelling. A minimum of two different conditions per medium was necessary for secondary modelling. From the 2319 kinetics and 2125 sets of fitted primary parameters, 2021 sets of primary growth parameters and 1232 kinetic data remained. A medium was defined by the nature of the medium and the source (publication, study, laboratory, ID record…). The growth rates were converted into growth rates derived from the Logistic model with delay, by using the conversion rates for the exponential, modified Gompertz, modified Logistic and Baranyi models of 1.00, 0.84, 0.86 and 0.97, respectively (Augustin et al., 2005). The database was divided into growth and inactivation data using intuitive criteria. At first, a linear regression was applied to the kinetic data. When the rate was positive, the bacterial behaviour was assimilated to growth, and when the rate was negative, the bacterial behaviour was assimilated to inactivation. Due to the complexity and the quality of the collected data, this criterion was compared to other criteria. The difference between the mean of the first three enumerations and the mean of the last three enumerations of each kinetic dataset was calculated. The kinetic data were then associated with growth when the difference was negative and with inactivation when the difference was positive. This criterion gave the same result as the rate for all kinetics. A visual inspection was used to check the result. 2021 sets of primary parameters and 697 kinetic datasets were associated with growth and 535 kinetic datasets with inactivation.

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Table 1 Product characteristics and storage conditions for the collected experiments. Number of data

T (°C)

pH

Sorbic acid (%)

Lactic acid (%)

aw

Type of data

Data source or reference

Media or food product

2 2 2 9 10 4 1 4 2 3 8 8 4 6 1 24 3

25 25 4 5–20 4–20 0–10 7 5–10 37 5–10 5–20 5–20 7 7 7 4 4–20

5.7 5.7 6.0 6.2 6.0–6.2 5.9 6.0 6.0 7.0 6.2 6.0 6.0 6.0 6.2 6.2 4.7–6.2 6.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.983 0.983 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.900 0.957–0.965 0.957–0.965 0.990 0.975 0.996 0.960–0.970 0.980

Kinetic μmax and/or lag Kinetic Kinetic μmax and/or lag Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or lag Kinetic Kinetic Kinetic Kinetic Kinetic

Beef Beef Beef Beef Beef Beef Beef Beef Beef Pork Pork Pork Pork Pork Pork Pork Pork

4

4–10

6.2

0

0

0.975

Kinetic

8 1 6 6 8 16 4 10 26 8 16 8 10 8 3 3 3 12 12 12 12 3 1 10 8 9 2 1 4 58

7 10 10–15 10–15 30–37 10–30 15 10–15 10–15 4–8 9–15 10–15 10–15 15–25 5–10 3–11 3–11 6–15 6–15 6–15 6–15 7 1 9–15 9–15 10–15 4 4 7 0–43

6.2 7.0 5.3 5.3 6.2 6.2 6.2 6.0–6.2 6.0–6.2 6.0–6.2 7.8 5.5 5.1 5.7 6.2 6.5 6.5 6.0 6.0 6.0 6.0 6.0 6.2 7.0 7.0 7.0 6.0 6.0 6.1 5.5–7.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.972 0.996 0.980 0.980 0.960 0.960 0.960 0.996 0.996 0.960 0.950 0.940 0.940 0.900 0.900 0.996 0.996 0.996 0.996 0.996 0.996 0.990 0.990 0.996 0.996 0.996 0.996 0.996 0.966 0.996

μmax and/or Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or Kinetic μmax and/or Kinetic μmax and/or Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or Kinetic μmax and/or

2 16 2 1 2 12 59 937 10 4 21 5 4 26 6 1 6 4 4 5 1 5 2

15 10–15 15 8 8 4–10 20 0–35 15–37 3–7 6 6 6 6 4–25 13 13 13 13 13 13 5–25 30

6.5 6.5 6.5 7.0 6.8 6.1 5.0–7.0 3.4–8.2 5.5 6.5 5.6–6.8 5.4–6.8 6.0 5.4–6.8 4.0–4.4 7.0 5.7–6.5 5.4–6.4 5.0–7.0 5.3–6.2 7.0 6.8 6.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0–0.2 0–2.6 0 0 0 0 0 0 2.1–2.3 0.1 0.0–0.1 0.5–2.7 0 0–2.4 0.03 0 0

0.950 0.950 0.950 0.996 0.970 0.957–0.996 0.996 0.996 0.950 0.996 0.996 0.996 0.930 0.996 0.812–0.996 0.996 0.995–0.997 0.965–0.99 0.996 0.996 0.996 0.971 0.996

Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or lag Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or lag Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic

Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Sym'previusa Burnett et al. (2005) Combasea Combasea Combasea Combasea Combasea Dykes (2003) Membré et al. (2004) Membré et al. (2004) Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Burnett et al. (2005) Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Sym'previusb Sym'previusb Sym'previusb Combasea Combasea Combasea Grau and Vanderlinde (1993) Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Unilever Unilever Combasea Combasea Combasea Combasea Combasea Combasea Lubem Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Sym'previusb Uhlich et al. (2006) Combasea

lag

lag lag lag

lag lag

Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Pork Poultry Poultry Poultry Poultry Poultry Poultry Poultry Poultry Poultry Poultry Poultry Poultry Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Other or unknown type of meat Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Cheese Milk (continued on next page)

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Table 1 (continued) Number of data

T (°C)

pH

Sorbic acid (%)

Lactic acid (%)

aw

Type of data

Data source or reference

Media or food product

1 2 5 2 2 1 4 5 2 2 2 3 2 7 5 5 2 3 2 24 13 14 18 8

4 30 4–30 35 35 10 10 10 32 32 10 10 7–13 7–13 10 10 4.2–10 0–12 4–13 4–40 4–35 4–35 4–35 4–16

6.6 6.6 6.6 6.4 6.4 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 5.1 7.0 7.0 6.0–7.0 7.0 7.0 5.4–7 6.5–6.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0 0 0 0 0 0

0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.957–0.965

Kinetic Kinetic μmax and/or Kinetic μmax and/or Kinetic Kinetic μmax and/or Kinetic μmax and/or Kinetic Kinetic Kinetic μmax and/or Kinetic μmax and/or Kinetic μmax and/or Kinetic μmax and/or μmax and/or μmax and/or μmax and/or Kinetic

Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Milk Other dairy product

8 10 24 1 1 2 2 2 3

10–15 10–15 10–20 20 5 5–20 5–20 5–20 5–10

4.6 7.0 4.6 7.0 7.0 7.0 6.2 6.2 6.4

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0.980 0.996 0.98 0.996 0.996 0.996 0.996 0.996 0.900

Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or lag kinetic μmax and/or lag Kinetic

2 2 11 2 2 4 9 18 3 7 3 4 2 15

5 5 4–20 5 5 2.9–12.8 10–15 10–15 10–25 5–20 5–10 5 5 4–12

6.8 6.0 6.2 6.3 5.7 5.7–6.2 7.1 6.7 6.8 6.8 4.8 5.8–6.0 7.2 3.7–5.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0–1.2 0 0 0 0 0 0 0 0

0.990 0.990 0.980 0.990 0.990 0.993 0.980 0.980 0.980 0.980 0.990 0.983–0.996 0.990 0.996

Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or lag

3 5

5–10 2–20

4.5 7.0

0 0

0 0

0.99 0.996

Kinetic Kinetic

10 10 1 1 4

15–37 15–37 22 22 2–12

7.1 4.3 6.6 6.6 6.0

0 0 0 0 0

0 0 0 0 0

0.770 0.750 0.996 0.996 0.996

Kinetic Kinetic μmax and/or lag Kinetic Kinetic

4

2–12

6.0

0

0

0.996

Kinetic

4 3 4 4 4 4 4 4 4 4 9

4–25 4–8 8–25 3–12 3–12 3–12 3–12 4–10 4–10 12–25 15–37

6.0–6.4 6.0 5.0–5.5 7.2 7.2 7.2 7.2 5.5–7.5 5.5–7.5 4.2–4.8 4.2

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0.–2.5 0

0.996 0.980 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.777 0.74

Kinetic μmax and/or lag Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic Kinetic

Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Combasea Sym'previusb Sym'previus(2) Sym'previusb Sym'previusb Sym'previusb Sym'previusb Gougouli et al. (2008) Sym'previusb Sym'previusb Sym'previusb Combasea Combasea Combasea Combasea Combasea Burnett et al. (2005) Combasea Combasea Combasea Combasea Combasea Combasea Sym'previusb Sym'previusb Sym'previusb Sym'previusb (Burnett et al., 2005) Combasea Combasea (Hwang and Tamplin, 2005) (Burnett et al., 2005) (Castillejo Rodríguez et al., 2000) Combasea Combasea Combasea Combasea Sinigaglia et al. (2006) Sinigaglia et al. (2006) Sym'previusb Sym'previusb Sym'previusb (Thomas et al., 1999) Thomas et al. (1999) Thomas et al. (1999) Thomas et al. (1999) Yoon et al. (2003) Yoon et al. (2003) Lubem Combase

10

15–37

4.7

0

0

0.800

Kinetic

Combase

lag lag

lag lag

lag lag lag lag lag lag lag

Other dairy product Other dairy product Other dairy product Egg product Egg product Egg product Egg product Egg product Fish Fish Fish Fish Fish fish Fish Fish Fish Fish Fish Crustacean Crustacean Crustacean Crustacean Vegetable or fruit Vegetable or fruit Vegetable Vegetable Vegetable Vegetable Vegetable

or or or or or

fruit fruit fruit fruit fruit

Vegetable or fruit Vegetable or fruit Vegetable or fruit Vegetable or fruit Vegetable or fruit Vegetable or fruit vegetable or fruit Vegetable or fruit Vegetable or fruit Vegetable or fruit Brine Other, mixed, uncategorized or unkown type of food Other, mixed, uncategorized or unkown type of food

L. Coroller et al. / International Journal of Food Microbiology 152 (2012) 139–152

143

Table 1 (continued) Number of data

T (°C)

pH

Sorbic acid (%)

Lactic acid (%)

aw

Type of data

Data source or reference

Media or food product

4 4 24 5

4–25 8–25 4–12 5–37

3.8–4.2 5.0–5.5 3.8–5.0 4.5

0 0 0 0

2.55 0 0 0

0.982 0.996 0.996 0.996

Kinetic Kinetic μmax and/or lag Kinetic

Salad dressing Sauce Sauce Culture medium

3

5–10

7.0

0

0

0.996

Kinetic

79 194 25 14 32

4–43 4–37 10–30 20–30 2–45

4.2–9.6 3.3–7.4 4.5–7.0 4.5–7.0 7.4

0 0 0 0 0

0 0 0 0 0

0.996 0.856–0.997 0.996–0.997 0.996–0.997 0.997

μmax and/or lag Kinetic Kinetic μmax and/or lag Kinetic

200

4–35

5.0–6.8

0–0.3

0

0.996

Kinetic

47 4 4 152 6 72 4 15 6 23 17 80 54 101 45 134 20 42 85 7 1 48 76 10 16 31 118 10 9 10 9 46 14 5 11 15 52 87 32 7 10 168 254 16

0.5–20 5 20 4–35 21 10–37 3.8–11.5 13 12 12–30 5–30 5–35 4–20 2–35 8–20 4–20 1–20 2–20 4–20 6–35 13 1–30 1–13 3–20 5–35 4–20 4–35 4 4–15 5–15 5–15 2–15 4–30 4–35 3–25 4–20 4–11.6 4–15 4–22 4–22 4–22 4–37 5–37 19–42

4.2–6.3 7.0 5.9–6.1 2.9–7.9 7.0 5.0–10.0 7.0 4.7–6.0 4.0–5.1 5.5–7.1 6.0–8.7 4.5–7.4 5.0–7.0 4.0–7.2 5.0–6.1 4.3–8.2 4.4–7.2 5.0–7.0 4.6–7.1 7.2–7.3 5.0 4.0–7.4 5.0–7.1 4.5–7.1 5.0–7.4 4.4–7.0 4.5–7.2 4.7–6.3 5.5–7.0 5.8–7.4 5.8–6.6 6.5–6.9 7.2 6.5 4.8–7.1 6.2 7.1 5.8–6.7 5.6–7.0 5.1–5.9 5–6.6.0 4.5–8.0 4.5–7.5 6.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1.5–2.0 0–18.0 0 0 0 0 0.0–0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.996 0.925–0.979 0.996 0.770–0.996 0.911–0.929 0.996 0.996 0.996 0.920–0.99 0.970–0.997 0.780–0.997 0.950–0.997 0.997 0.997 0.997 0.997 0.997 0.954–0.986 0.961–0.997 0.997 0.997 0.935–0.997 0.915–0.997 0.915–0.997 0.947–0.997 0.950–0.997 0.911–0.999 0.960–0.974 0.949–0.997 0.987–0.995 0.992–0.995 0.999 0.990 0.996 0.940–0.986 0.980 0.997 0.955–0.998 0.933–0.994 0.993–0.994 0.964–0.997 0.964–0.997 0.950–0.997 0.996

Kinetic Kinetic Kinetic Kinetic Kinetic μmax and/or Kinetic Kinetic Kinetic μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or μmax and/or

Lubem Sym'previus Sym'previus Buchanan and Klawitter (1990) Burnett et al. (2005) Chawla et al. (1996) Combase Combase Combase Yeu-Hsin and Schaffner (1993) (El-Shenawy and Marth, 1989) Combasea Combasea Combasea Lubem Combasea Combasea Sym'previus Sym'previus Sym'previus Sym'previus Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Unilever Zaika and Fanelli (2003)

lag

lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag lag

Culture medium Culture Culture Culture Culture Culture

medium medium medium medium medium

Culture medium Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture Culture

medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium medium

a Searching criterion in Combase : All type of food, Listeria, aerobic conditions, all preparation, all additive, all other conditions and without microflora (web site query 20 February 2009). b Searching criterion in Sym'previus: All type of food, Listeria, all factors (web site query 20 February 2009).

2.2. Predictive models 2.2.1. Concept The global model is based on the gamma concept (Zwietering et al., 1992), which allows the quantification of the individual effects of each explanatory factor independently of the observed bacterial behaviour. It can be formulated as follows for bacterial growth: k

μ max ¼ μ opt : ∏ γXi ðXi Þ i¼1

ð3Þ

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Where μmax is the maximal growth rate in given environmental conditions (Xi), the γXi function reflects the effect of each environmental factor (Xi) on bacterial growth. The synergy or antagonism of the factors on bacterial growth can be taken as a single factor effect by an additional γXi function. The γ value varies from 0 to 1. If the γ value of a given factor or interaction is 0, growth is prevented. If the γ value is equal to 1, growth potential is optimal for the considered factor (Le Marc et al., 2002). μ opt is the growth rate in optimal conditions for a given medium and a given microorganism. 2.2.2. Growth modelling 2.2.2.1. Primary modelling. The Logistic model with delay (Rosso, 1995) was used as the primary model. Its form is given in Eq. (4) below.

lnðNÞ ¼

8 <


lnðN0 Þ 
 if t≤lag N max −μ max ·ðt−lag Þ −1 ·e if tNlag : lnðN max Þ− ln 1 þ N0

ð4Þ

Where N is the bacterial concentration at a given time (t), Nmax is the maximal bacterial concentration; N0 is the initial bacterial concentration; the bacterial concentration unit was CFU/mL or CFU/g depending on the nature of the media liquid or solid respectively; μmax (h − 1) is the bacterial specific growth rate; and lag (h) is the latency time before growth. As the assumption of a proportional relation between μmax and lag was made (Augustin and Carlier, 2000), the ln(K) value was estimated using Eq. (5). lnðK Þ ¼ lnðμ max ·lag Þ

ð5Þ

2.2.2.2. Secondary modelling. The collected or estimated growth rates (μmax) from Eq. (4) were modelled following Eq. (3). Six secondary models were tested. These models differed by the γ-terms for each factor and also by whether an interaction term was added or not. 2.2.2.2.1. Models without an interaction term. Cardinal model. The cardinal model (Rosso et al., 1995) was used to quantify the influence of temperature, pH and water activity on bacterial growth. Each γ function is written as follows: 8 > > > <

0 if X ≤X min ðXi −X max Þ·ðXi −X min Þn n−1         if X ibX bX γXi ðXi Þ ¼  min i max > Xopt −X min · Xopt −Xi min · Xi −Xopt − Xopt −X max · ðn−1Þ:Xopt þ X min −n:Xi > if Xi ≥X max > : 0

ð6Þ

where Xi is an environmental factor, Xmin, Xopt, and Xmax are the minimal, optimal and maximal values of the factor for bacterial growth. The n value was 2 for temperature and 1 for pH and water activity (Couvert et al., 2010). For the acid influence on the bacterial growth the following model was chosen:

γAH ðAH Þ ¼


 ½AH  α 1− MIC

ð7Þ

Here, [AH] is the undissociated concentration of the considered acid, α is a shape parameter and MIC is the minimum inhibitory concentration. The α value was fixed at 0.3 and 1 for sorbic and lactic acid respectively (Zuliani et al., 2007). Following this model, the effect of a mix of acids γAH([AHtot]) can be written as follows: γAH ðAHtot Þ ¼ γsorbic ð½AHsorbic Þ:γlactic ð½AHlacict Þ

ð8Þ

Where [AHsorbic] and [AHlactic] represented the undissociated concentration of sorbic and lactic acids respectively. Lambert model. Assuming no-interaction between the environmental factors, Lambert and Bidlas proposed another approach to model the growth rate (Lambert and Bidlas, 2007). The model was applied to quantify the influence of temperature, pH, aw and the two acids on the specific growth rate. pffiffiffiffiffiffiffiffiffiffiffi μ max ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ opt ·γT ðT Þ·e−ðγpH þγaw þγsorbic þγlactic Þ

ð9Þ

For temperature, the classical cardinal model was used (Eq. (6)), and for the other factors, it can be written as :
γxi ðXi Þ ¼

½Xi  P2i−1

p

2i

ð10Þ

where Xi the studied factor, p2i the slope parameter and p2i−1 the concentration of the antimicrobial substance which results in an inhibition of the optimal growth rate of 1/e. For pH, the proton concentration was taken into account. For the acids, the undissociated acid concentration was taken into account. For the water activity, the sodium chloride concentration was used but the notation aw was used to ensure consistency. Using this model, the first step was to calculate the EffC criterion as follows (Bidlas and Lambert, 2008; Lambert and Lambert, 2003):  p Q EffC ¼ γpH þ γaw þ γsorbic þ γlactic

ð11Þ

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145

This criterion is useful for combined inhibitors. To define the growth-inactivation boundary, it was used as follows. Three cases are possible: i) no inhibitory substance is present in the environment: h i þ H þ ½NaCl þ ½AHsorbic  þ ½AHlactic  ¼ 0

ð12Þ

Therefore the growth rate μmax is estimated as follows: pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ max ¼ μ opt ·γT ðT Þ

ð13Þ

ii) if log (EffC) is negative, the growth rate is estimated as follows: pffiffiffiffiffiffiffiffiffiffiffi μ max ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pQ μ opt ·γT ðT Þ·e−ðγpH þγaw þγsorbic þγlactic Þ

ð14Þ

iii) or else, the growth rate is estimated as follows: pffiffiffiffiffiffiffiffiffiffiffi μ max ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  μ opt ·γT ðT Þ· · 1−pQ · logðEffC Þ e

ð15Þ

In this case the pQ parameter value equals 2. 2.2.2.2.2. Models with an interaction term. The other secondary growth models utilised in our study take into account the interaction between factors (see Eq. (16) below): pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ max ¼ μ opt ·∏γXi ðXi Þ·ξ

ð16Þ

Where ξ is the interaction between factors. Separate effect of the weak acids model. According to the paper by Le Marc et al. (2002), the growth rate can be modelled as: pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ max ¼ μ opt ·γT ðT Þ·γpH ðpHÞ·γaw ðaw Þ·γAH ð½AHsorbic ; ½AHlactic Þ·ξ

ð17Þ

with γAH([AHsorbic], [AHlactic]) written as in Eq. (8). Eq. (6) was used to reflect the influence of temperature, pH, and water activity on the bacterial growth rate, and Eq. (7) to describe the acid inhibitory effect. For each environmental factor, a φ value can be calculated as follows:  pffiffiffiffiffiffiffiffiffiffiffiffi2 φðT Þ ¼ 1− γT ðT Þ

ð18Þ

 2 φðpHÞ ¼ 1−γpH ðpHÞ

ð19Þ

 3 φðAwÞ ¼ 1−γaw ðaw Þ

ð20Þ 2

φð½AHsorbic Þ ¼ ð1−γAH ð½AHsorbic ÞÞ

ð21Þ

2

ð22Þ

φð½AHlactic Þ ¼ ð1−γAH ð½AHlactic ÞÞ

Then, a ψ value is calculated as follows: ψ¼∑ i

φðXi Þ    2 ∏ 1−φ Xj

ð23Þ

j≠i

and the ξ value is : 8 ; ψ≤0:5 <1 ξ ¼ 2 ð1−ψÞ ; 0:5bψb1 : 0 ; ψN1

ð24Þ

Multiplicative effect of the weak acids model. The influence of the weak acids can be modelled in different ways. For instance, Coroller et al. (2005) proposed a multiplicative effect of the weak acids model, incorporated in a φ term as follows: 2

φð½AHsorbic ; ½AHlactic Þ ¼ ð1−γsorbic ðAHÞ×γlactic ðAHÞÞ

ð25Þ

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Dalgaard model. Dalgaard's model (Mejlholm and Dalgaard, 2007) uses the interaction model of Le Marc et al. (2002) as described in Eq. (17), but it differs in the functions of the γ terms for temperature, pH and water activity. These functions are written as follows: 8 > <

T−T min γT ðT Þ ¼ T > opt −T min : 0 γpH ðpHÞ ¼

!2 if T N T min

ð26Þ

if T ≤ T min

pH min −pH

if pH N pH min if pH ≤ pH min

1−10

0

8 < aw −aw min γaw ðaw Þ ¼ awopt −aw min : 0

ð27Þ

if aw N aw min

ð28Þ

if aw ≤ aw min

For the influence of the acids, the same γ term as those suggested by Le Marc et al. (2002) were kept (Eqs. (7) and (8)). Likewise, the expressions of the φ value are the same as those defined by Le Marc et al. (2002), as it can be seen Eqs. (18)–(22). Augustin model. With the same definition of the γ terms (Eqs. (6) and (7)) and interaction term (Eqs. (23) and (24)) as Le Marc et al. (2002), the Augustin model differs in the mathematical expression of the φ term (Augustin et al., 2005; Zuliani et al., 2007). 3

φðT Þ ¼ ð1−γT ðT ÞÞ

ð29Þ

 3 φðpHÞ ¼ 1−γpH ðpHÞ

ð30Þ

 3 φðaw Þ ¼ 1−γaw ðaw Þ

ð31Þ 3

φðAHÞ ¼ ð1−γAH ð½AHÞÞ

ð32Þ

2.2.3. Inactivation modelling 2.2.3.1. Primary modelling. The bacterial inactivation kinetics were fitted from the double Weibull model (Coroller et al., 2006) :
log10 ðNÞ ¼ log10

N0 1 þ 10α

 p

 þ log10 10

−

t δ1

þα

 p !

−

þ 10

t δ2

ð33Þ

Where N0 (CFU/mL) is the initial bacterial concentration; δ1 (h) is the time needed for the first decimal reduction of a sensitive subpopulation 1; δ2 (h) is the time needed for the first decimal reduction of a more resistant subpopulation 2; p is a shape parameter; and α is the logit of the fraction of subpopulation 1 within the population. This therefore implies that log 10(N0) ≥ α and δ1 ≤ δ2. The advantages of the double Weibull model are discussed by Coroller et al. (2006). This model is able to describe complex curve shapes (sigmoid, biphasic, convex) and simpler ones (concave, and linear). A common p value was estimated on all inactivation curves. In simpler cases (identified using the AIC criterion, see Section 2.4), the model was simplified into a Weibull model (Mafart et al., 2002). For example when the stress is weak, the resistance of each subpopulation cannot be distinguished, then δ1 could be assumed to be equal to δ2·(δ1 = δ2 = δ in Eq. (34)). This might occur as well if the population is homogenous, i.e. only subpopulation 1 or 2 is observed.
p t log10 ðNÞ ¼ log10 ðN0 Þ− δ

ð34Þ

2.2.3.2. Secondary modelling. The gamma concept was extended to bacterial resistance for the considered factors using Eq. (35) (Mafart, 2000): 1 1 ¼ ·∏λXi ðXi Þ δ δ

ð35Þ

it is equivalent to: 

log10 δ ¼ log10 δ −∑ log10 λXi ðXi Þ

ð36Þ

This equation was used to fit bacterial resistance. The mathematical expressions of the λ terms are provided in the following paragraph. Note that for the temperature inhibitory effect, two secondary models, namely models A and B, were tested. Temperature, model A. The λ term for temperature influence could be written as follows (Coroller, 2006):   2: Tc −Topt   · T−T 2 Z T    log10 λT ðT Þ ¼
 > > Tc −Topt · 2T  −Tc −Topt T−Topt 2 > > > − : ZT ZT2 8 > > > > > <

T≤Tc ð37Þ TNTc

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147

where Tc is a connection point between the linear function and the hyperbola in degrees Celsius; Topt is the optimal temperature for survival observed in the case of strong acid stress; T* is a reference temperature arbitrarily fixed at 12 °C, and zT the variation in temperature from Topt which gives a decimal variation of the bacterial resistance. Temperature, model B. The log-linear evolution of the inactivation rate as a function of the storage temperature has been demonstrated for Escherichia coli and L. monocytogenes (McQuestin et al., 2009; Zhang et al., 2010). A second model (inactivation model B) has been tested for the storage temperature, it can be written as follows: log10 λT ðT Þ ¼

ðT−T  Þ ZT

ð38Þ

pH. For pH, the λi term could be written as proposed by Coroller (2006): log10 λpH ðpHÞ ¼

pH−pHopt ZpH

!2 ð39Þ

With the same interpretation of the parameters pHopt and zpH as with the temperature factor. The pHopt was fixed at 6.5. aw. The λi term was also determined for aw: log10 λaw ðaw Þ ¼

aw −awopt

!2 ð40Þ

Zaw

Where aw opt is equal to 0.996. Acids. For lactic and sorbic acids, the λi term was calculated using Eqs. (41) and (42) respectively: log10 λlactic ð½AHlactic Þ ¼

log10 λsorbic ð½AHsorbic Þ ¼

½AHlactic  ZAH lactic

!1

½AHsorbic  ZAH sorbic

2

ð41Þ ! ð42Þ

2.3. Parameters estimation In order to estimate parameters, the following process was implemented. The cases in which growth was observed were used to fit the primary growth models. Then, the secondary growth model was fitted to the estimated or collected growth rates. In the same way, the inactivation data were used for the primary and secondary modelling. The primary and secondary models can be expressed as follows: Yi = f(xi, θ) + εi. Where Yi is the observed response. In the case of growth, Yi was the natural logarithm of N (CFU/mL) or the square root of the growth rate. In the case of inactivation, Yi was the decimal logarithm of N (CFU/mL), or the decimal logarithm of δ (h). f is the regression function for the primary or secondary model. xi is the explanatory factor, in our case time or temperature, pH, water activity and undissociated acid concentrations. Vectors of parameters of models θ were estimated by minimization of the sum of squares of the residual values (εi) defined by: n

2

C ðθÞ ¼ ∑ ðYi −f ðxi ; θÞÞ

ð43Þ

i¼1

Where n is the number of data. The minimum C(θ) values were computed by non-linear fitting module (lsqcurvefit or fmincon if constraints are needed, MATLAB 2009b, Optimization toolbox, The Math-works). Confidence intervals of the estimated parameters were obtained by linear approximation (nlparci, MATLAB 2009b, The Math-works). 2.4. Model performance evaluation The models were evaluated according to three criteria: practicality, quality of prediction of the growth/inactivation boundary and quality of fit of the secondary model. 2.4.1. Practicality When assessing the validity of the proposed models, one can compare the estimated secondary parameters to some ‘generally accepted’ reference values. Such reference values (e.g. for Tmin or pHmin) are often published in the literature. They usually have a biological meaning, so they can also be obtained from expert opinion. 2.4.2. Growth/no-growth boundary prediction The evaluation of the growth-inactivation boundary was made by assessing the ability of models to predict the Listeria spp. behaviour correctly (growth or inactivation). To do so, the product of the γ-terms in Eqs. (3), (9) or (16) was calculated. If this value was strictly positive, bacterial growth was expected and growth models were therefore utilised. On the other hand, if the product of the γ-terms was equal to

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zero, then bacterial inactivation was assumed and consequently, non-thermal inactivation models were utilised. Subsequently, for each model, five criteria were computed (Augustin et al., 2005; Hajmeer and Basheer, 2003): FPN percentage of cases that were correctly predicted FP percentage of cases for which growth is predicted and inactivation occurs FN percentage of cases for which inactivation is predicted and growth occurs PPV proportion of cases for which growth is predicted and growth occurs NPV proportion of cases for which inactivation is predicted and inactivation occurs These criteria were computed by simulating the microbial behaviour as described above and comparing the results with the observed data. 2.4.3. Goodness of fit The RMSE was computed to reflect how the models fit the data. As the mean error was strongly influenced by extreme values, the median error of the prediction was also computed. This criterion was used in two ways. At first, it was computed from the secondary model fit, comparing growth rates (or delta values for inactivation) calculated using a secondary model with those generated from the kinetics. Secondly, the RMSE criterion was computed to reflect the global error made by the primary and the secondary models. This global error was called prediction error. In this case, calculations were made only in cases where growth was predicted and occurred, and where inactivation was predicted and occurred. The simulations of log10 N were carried out with a two-step modelling procedure involving primary models (predicting kinetics versus time) and subsequently secondary models (predicting growth or inactivation rate parameters versus temperature and formulation factors). The primary parameters, N0, Nmax and α were fixed for performing these simulations. The Akaike Information Criterion (AIC) was also computed to compare models on the goodness of fit and the parsimony (Akaike, 1973; McQuarrie and Tsai, 1998). A great number of parameters or a poor quality of fit (small log-likelihood value) corresponds to a high AIC value. The best model has the lowest AIC value.
 C ðθÞ AIC ¼ n· log þ n· logð2πÞ þ n þ 2·ðp þ 1Þ n

ð43Þ

Herein p is the number of parameters of the model, and n the number of observation. The AIC was computed for both growth and inactivation models. Moreover, in the case of the primary inactivation model, the simplification of the double Weibull into a Weibull model was implemented on the basis of the AIC criterion: when the Weibull model was associated with a lower AIC compare to the double Weibull model, the simplification was performed.

3. Results The 697 growth kinetic datasets were individually fitted by the Eq. (4). Growth rates, μmax, were retained individually to be used as statistical responses in a secondary modelling phase. In addition, a unique ln(K) was retained for calculating log10 N as a function of time, storage and formulation conditions when combining primary and secondary models. The ln(K) estimated value was 1.37 ± 0.37. This simplification (one ln(K) value instead of 697 values) slightly increased the RMSE of the curve fitting by a mean value of 5.86% (median value of 0.95%) giving an average final RMSE value of 0.511. The effect of five explanatory factors—namely pH, aw, sorbic and lactic acid (product formulation) and storage temperature—on the growth/no-growth interface was studied. Six secondary growth models were applied to calculate wheter either growth or inactivation was expected taking these five factors into account. Three secondary models had a classification rate (i.e. the percentage FPN of correct predictions) over 85% (Table 2). The Lambert model gave a FPN value below 65%, and in 38% of cases it predicted growth while inactivation occurred. Close to the growth/no-growth boundary (γ = 0.05, Fig. 1), this latter model failed to predict the cases in which Table 2 Evaluation criteria of the growth/inactivation boundary. Secondary growth model

FPN (%)

FP (%)

FN (%)

PPV (%)

NPV (%)

Cardinal model Lambert model Separate effect of the weak acids model Multiplicative effect of the weak acids model Augustin model Dalgaard model

80.58 61.98 84.96 85.05 87.06 85.30

18.63 37.78 13.76 13.70 10.99 13.12

0.79 0.24 1.28 1.25 1.95 1.58

98.71 99.60 97.91 97.96 96.81 97.41

52.04 2.74 64.58 64.73 71.71 66.22

inactivation occurred (FPN b 10%, Fig. 1) resulting in only a few cases in which inactivation is predicted and inactivation occurs (NPV = 3%, Table 2). On the other hand, with the data collected for Listeria spp. for the five explanatory factors considered in this study, there was no substantial difference between the models in terms of the percentage of cases in which growth occurred when it was predicted (PPV ranging from 96.8 to 99.6%, Table 2). Likewise, there was no difference among the FPN values calculated using the six models when γ was around 0.15 or 0.20 (Fig. 1). This lack of difference is likely to be due to the absence of data in this slow growth area. In a second step, the ability of the secondary growth and inactivation models to predict growth or inactivation rates, respectively, was assessed. For the growth area, the six secondary models gave very similar results (Table 3), with the AIC ranging from − 1818 to −1525, the RMSE (μmax) from 0.22 to 0.24 and the median log10N error from 0.31 to 0.34. Moreover, the log10N relative error appeared to be very similar (Fig. 2) whatever the model used. The estimated parameters Tmin, pHmin, aw min, MICsorbic and MIClactic values were of the same order of magnitude, apart from the Lambert model. Indeed, the Lambert model gave higher pHmin and aw min than those obtained with the secondary models and lower MIC values (Table 4). This observation was expected due to the definition of these values as the concentration of the inhibitory substance which gives a 1/e fold inhibition of the optimal growth rate (see Eq. (10)). The optimal growth rate (μopt) and the average prediction error per dataset (i.e. per experimental plan, per food, per source) are provided in Tables 1 and 2 of the supplementary materials. The variability in μopt is high, with for instance values from 0.182 up to 2.024 h − 1 among the dataset collected from culture media (mean value of 0.495 ± 0.320 h− 1). This variability was observed for each food class, and it was difficult to define a growth trend (expressed through the μopt value) for

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149

Fig. 2. Distribution of the relative error of the log10N predicted values for each combination of the primary model (Logistic model with delay) with one of the six secondary growth models. Each bar represents a secondary model with, from left (dark bar) to right (white bar), successively the Cardinal model, Separate effect of the weak acids model, Multiplicative effect of the weak acids model, Augustin model, Dalgaard model, Lambert model.

Fig. 1. Percentage of cases correctly predicted (FPN) versus various γ values for each secondary growth model. For each γ value, each bar represents a secondary model, with from left (dark bar) to right (white bar), successively the Cardinal model, Separate effect of the weak acids model, Multiplicative effect of the weak acids model, Augustin model, Dalgaard model, Lambert model.

a given class of products, even if higher values of μopt were generally observed for vegetables or fruits (1.109 ± 0.351 h− 1) and seafood products (3.614 ± 5.710 h− 1). For dairy products, the mean of the μopt value was 0.585 ± 0.498 h− 1, with a mean value of 0.510 ± 0.265 h− 1 for cheese and 0.367 ± 0.086 h− 1 for milk. For meat products, the mean value was 0.634 ± 0.737 h− 1, with a mean value of 0.187 ± 0.145 h− 1 for beef and 0.765 ± 0.830 h− 1 for poultry. In terms of model evaluation, in cases in which growth was predicted and occurred, no model could be related to a class of product, based upon its goodness of fit (Table 2, supplementary materials). The 535 inactivation kinetics were individually fitted by Eqs. (33) or (34) with the same shape parameter (p). This unique p value was estimated at 1.99 ± 0.16. The double Weibull model (Eq. (33)) was fitted on 407 kinetics, and simplification in a single Weibull model (Eq. (34)) was implemented, based upon the AIC criterion, in only 128 cases. To describe the effects of pH, aw, sorbic and lactic acids, only one secondary model was utilised for each term, whilst for the temperature effect, two secondary inactivation models were compared. For these two temperature models, no substantial difference in terms of goodness of fit (AIC, RMSE criteria) or log10N relative error was observed (Tables 5, 6 and Fig. 3). In contrast, the estimated z parameters changed significantly when the resistance δ was calculated using either the nonlinear model (Eq. (37)) or the loglinear model for temperature (Eq. (38)).The ZT value shifted from 17 (Table 5) up to 70 (Table 6), indicating a very low effect of storage temperature Table 3 Evaluation criteria of the quality of the secondary growth model. Secondary growth model

AIC

RMSE (μmax)

median error (log10N)

Cardinal model Lambert model Separate effect of the weak acids model Multiplicative effect of the weak acids model Augustin model Dalgaard model

− 1593 − 1818 − 1574 − 1648 − 1625 − 1525

0.0229 0.0238 0.0231 0.0223 0.0222 0.0230

0.313 0.341 0.320 0.317 0.310 0.329

on the inactivation rate in the case of predictions made using the simplified secondary temperature model (Eq. (38)). The reference resistance (δ*), the range of alpha values, and the average prediction error per food are provided in Tables 3 and 4 of the supplementary materials. The δ* values ranged from a few days (close to one week for beef) to a few months (close to two months for a cheese). Like for growth, no inactivation behaviour trend can be identified for a given type of food from our results. The meat and cheese products were represented througthout the entire range of δ* values. The global error for the predicted microbial population size was less than 0.71 log10(CFU/mL) with a median value of less than 0.21 for all media. Overall, the prediction error in the food matrices was lower than in culture media, but the amount of data and the range of the tested conditions wider. Nevertheless, the prediction error in food products was quite low for cases in which inactivation was predicted and occurred with log10N values from 0.097 to 0.101 (crustacean products) to 0.304–0.305 (cheese). The model system combining growth/no-growth interface followed by inactivation or growth rate predictions can be utilised to calculate the expected log10 N as a function of time, storage and formulation conditions (Fig. 4). A cheese product was choosen as an example (see supplementary material). The μopt was equal to 0.5 h − 1 and the log10δ* values were equal to 3.5 and 3.6 for the most sensitive and the most resistant populations respectively. After one day at ambient temperature (20 °C), neither growth or inactivation occurred at a pH value between 3.5 and 6.0 (Fig. 4a). In contrast, Listeria is predicted to achieve up to a 5 log10 increase after 1 week at 20 °C and pH 5.5 (Fig. 4b), or even 8 log10 in the same conditions for 1 month (Fig. 4c). On the other hand, if the pH is lowered to 4.5, at 20 °C, Listeria growth is inhibited (Fig. 4b). To obtain at least 1 log10 reduction after one month at refrigeration temperatures (b8 °C), the pH has to be lower than 4.6 (Fig. 4c). 4. Discussion A single modelling approach covering inactivation, growth/nogrowth and growth of pathogen or spoilage bacteria is valuable since it enables the prediction of bacterial behaviour whatever the product storage or formulation conditions, without either discontinuity or contradictory predictions. For example with the Listeria dataset, after one week at ambient temperature (20 °C), when increasing the pH from 3.5 up to 5.5, the bacterial behaviour is predicted to change gradually from a reduction (− 2 log10) up to an increase (+5 log10) (Fig. 4b). Consequently, the model can be used to quantify the increase or decrease of the bacterial population for a given formulation

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Table 4 Estimated values of the secondary parameters of growth and their associated standard deviation. Secondary growth model

Tmin (°C)

Cardinal model Lambert model

− 6.02 ± 1.47 − 5.63 ± 0.63

Separate effect of the weak acids model Multiplicative effect of the weak acids model Augustin model Dalgaard model

− 6.57 ± 1.17 − 5.90 ± 1.10 − 5.82 ± 0.86 − 8.47 ± 1.38

pHmin

aw

4.03 ± 0.08 4.57 ± 0.02⁎ 1.95 ± 0.39⁎⁎ 3.93 ± 0.06 3.90 ± 0.06 4.03 ± 0.05 4.37 ± 0.01

min

0.90 ± 0.01 0.94⁎⁎⁎ 1.94 ± 0.18⁎⁎ 0.90 ± 0.01 0.90 ± 0.01 0.90 ± 0.01 0.89 ± 0.01

MIC (mM) Sorbic acid

Lactic acid

9.50 ± 1.60 2.55 ± 0.33⁎ 0.99 ± 0.42⁎⁎ 13.19 ± 1.69 13.10 ± 1.64 10.33 ± 1.45 10.52 ± 1.55

7.14 ± 1.17 6.13 ± 2.72⁎ 0.47 ± 0.19⁎⁎ 7.96 ± 0.56 7.9 ± 0.53 6.92 ± 1.07 6.52 ± 0.74

⁎ p2i − 1 parameter. ⁎⁎ p2i parameter. ⁎⁎⁎ The estimated value was 9.89 ± 0.79% NaCl equivalent to an aw value of 0.94.

Table 5 Estimated values of the secondary inactivation parameters using temperature model A (Eq. (37)) after the fitting of 535 kinetics using the double Weibull model (Eq. (33)) or the Weibull model (Eq. (34)). Parameters

Estimated value for δ1

Estimated value for δ2

ZpH ZAH sorbic (mM) ZAH lactic (mM) ZNaCl (%) ZT Tc RMSE AIC

2.79 ± 0.07 84.87 ± 100.92 49.57 ± 5.19 9.36 ± 0.88 16.66 ± 0.43 19.87 ± 4.74 0.406 592

2.63 ± 0.10 21.9 ± 10.81 60.28 ± 11.2 12.24 ± 0.91 17.41 ± 0.83 20.38.09 0.510 836

or storage condition. Moreover, the model can be utilised the other way around as it enables the optimization of a food formulation or a storage condition for a given targeted increase or decrease of the bacterial population, i.e. to comply with a pre-defined Performance Criteria (FAO/WHO, 2002). Indeed, the continuity in predictions offers the possibility of a complete mapping of the product formulation space for a given microbial behaviour. The initial number of formulation solutions might then be reduced due to the formulation constraint and/or the food technology properties to achieve a satisfactory product formulation. The tiered approach developed in this study also takes time into account. Fig. 4 shows that if the product is stored for 1 week at a refrigerated temperature under 10 °C, no log-increase should be observed whatever the pH. If the product is stored for 1 month, the storage temperature should be under 5 °C whatever the pH to fulfil the no-log increase criterion. Using the growth boundary model, no log increase should be observed for a pH of less than 5.2 and a storage temperature lower than 5 °C or a pH less than 4.6 and a storage temperature lower than 1 °C. Incorporating the time factor increases the dimensions of the formulation space and therefore the flexibility for the development of new products. The model may be then utilised by microbiological food safety experts or even to be incorporated in a research and development predictive toolbox (Membré and Lambert, 2008).

In this study, six secondary models were computed and compared for their ability to discriminate between growth and no-growth areas and to subsequently predict the growth behaviour of Listeria spp. With the data collected from Listeria spp., the secondary models ‘Multiplicative effect of the weak acids’, ‘Augustin model’ and ‘Dalgaard model’ gave results which were of the same order of magnitude, their accuracy was acceptable with more than 85% of growth/no-growth cases correctly predicted and a RMSE (log10 N) below 0.5 in both growth and inactivation areas. These three models may be recommended when performing a similar modelling approach with another set of data (other explanatory factors) or with another microorganism. This may be the case in the field of microbial spoilage where slight growth is accepted. When building a single modelling approach, the advantage of using secondary growth models based upon the γ-concept lies in their capacity to predict the growth/no-growth interface, without adding any supplementary estimated parameters. In the literature, the alternative often adopted is to build two distinct models, one model for the growth area, another one for the growth/no-growth interface area. Often, the latter model is developed using logistic regression (Panagou et al., 2010; Presser et al., 1998; Vermeulen et al., 2009) for which many experimental data are required. In comparison, a modelling approach, based upon the γ-concept, is not data demanding, which is both more time and cost efficient. Another advantage of using a growth/no-growth interface model based upon the γconcept is the possibility of obtaining rough model estimates by expert elicitation, as the cardinal values are relatively close to the values known by food microbiologists as critical points, for instance Tmin relatively close to the temperature value below which no growth is

Table 6 Estimated values of the secondary inactivation parameters using temperature model B (Eq. (38)) after the fitting of 535 kinetics using the double Weibull model (Eq. (33)) or the Weibull model (Eq. (34)). Parameters

Estimated value for δ1

Estimated value for δ2

ZpH ZAH sorbic (mM) ZAH lactic (mM) ZNaCl (%) ZT RMSE AIC

2.80 ± 0.08 55.7 ± 45.87 49.76 ± 5.52 9.79 ± 0.10 69.72 ± 14.08 0.420 626

2.65 ± 0.1 20.23 ± 9.33 61.09 ± 11.56 12.21 ± 2.02 68.69 ± 21.06 0.646 1089

Fig. 3. Distribution of the relative error of the log10N predicted values for the primary models (Weibull models) with the two secondary inactivation models. The black bar represents the non-linear temperature model (Eq. (37)), and the white bar the loglinear temperature model (Eq. (38)). For both secondary models, 3% of the relative error was outside of the range [− 2;2].

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151

Fig. 4. Log-variation of the population size of Listeria using the multiplicative effect of the weak acids model as the secondary growth model and the temperature model A as the secondary inactivation model. In a product with 0.98 water activity and 0.2% lactic acid concentration, the chosen μopt was equal to 0.5 and the log10δ* values equal to 3.5 and 3.6 for the more sensitive and the more resistant population respectively. The ratio between these two subpopulations was equal to 10 (α = 1). The storage time was (a) 1 day, (b) 1 week and (c) 1 month.

expected. This might be helpful when a model has to be developed quickly to provide an estimation of microbial behaviour. However, among the six models tested, and again with the dataset used in this study, some of them seemed inappropriate for such usage. For instance, the Dalgaard model gave a Tmin value much lower (8 °C lower) than the minimum growth temperatures commonly accepted for L. monocytogenes (ICMSF, 1998). This may be explained by the mathematical expression used to describe the inhibitory effect of the temperature and/or the specific data set used in this study. Indeed, in our data collection, there were some high growth rates reported at low temperatures in dairy, meat, seafood products and synthetic media, and this could explain such a low minimal temperatures of growth and high estimated values of μopt (Supplementary material, Table 1). Nevertheless, no general conclusion on the secondary model cardinal parameters can be drawn from our study since other authors have reported Tmin values of −3 °C for L. monocytogenes using the Augustin model and/or the Dalgaard model (Augustin et al., 2005; Mejlholm and Dalgaard, 2007). The modelling approach was set up with six secondary growth models but only two secondary inactivation models (and these two inactivation models varied only in their temperature effect expression), which may appear unbalanced. This is due to the fact that in the literature, fewer secondary models have been developed to describe non-thermal inactivation compared to those developed for growth; hopefully more non-thermal inactivation models will be suggested in the future. With the dataset used, it is difficult to make definitive conclusions on the two secondary temperature inactivation models investigated in this study. In conclusion, a modelling approach covering growth, growth/nogrowth and inactivation areas has been developed using a Listeria. spp. dataset. The model can be used to predict the bacterial behaviour whatever the formulation condition or conversely, to design a food product for a given targeted log count. It would be interesting to investigate whether a similar tiered-modelling approach could be applied to other microorganisms, in the field of food safety or food stability. It could be applied to single use or multi-use products that may become contaminated after opening. Supplementary materials related to this article can be found online at doi:10.1016/j.ijfoodmicro.2011.09.023. Acknowledgements We gratefully acknowledge Peter McClure for his interesting comments and valuable suggestions towards the manuscript. These results were presented at the 22nd International ICFMH Symposium, Food Micro 2010, 30 August–3 September 2010, Copenhagen, Denmark.

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