Structural model requirements to describe microbial inactivation during a mild heat treatment

International Journal of Food Microbiology 59 (2000) 185–209 www.elsevier.nl / locate / ijfoodmicro

Structural model requirements to describe microbial inactivation during a mild heat treatment A.H. Geeraerd, C.H. Herremans, J.F. Van Impe* BioTeC-Bioprocess Technology and Control, Department of Food and Microbial Technology, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 92, B-3001 Leuven, Belgium Received 17 August 1999; received in revised form 1 May 2000; accepted 27 May 2000

Abstract The classical concept of D and z values, established for sterilisation processes, is unable to deal with the typical non-loglinear behaviour of survivor curves occurring during the mild heat treatment of sous vide or cook–chill food products. Structural model requirements are formulated, eliminating immediately some candidate model types. Promising modelling approaches are thoroughly analysed and, if applicable, adapted to the specific needs: two models developed by Casolari (1988), the inactivation model of Sapru et al. (1992), the model of Whiting (1993), the Baranyi and Roberts growth model (1994), the model of Chiruta et al. (1997), the model of Daughtry et al. (1997) and the model of Xiong et al. (1999). A range of experimental data of Bacillus cereus, Yersinia enterocolitica, Escherichia coli O157:H7, Listeria monocytogenes and Lactobacillus sake are used to illustrate the different models’ performances. Moreover, a novel modelling approach is developed, fulfilling all formulated structural model requirements, and based on a careful analysis of literature knowledge of the shoulder and tailing phenomenon. Although a thorough insight in the occurrence of shoulders and tails is still lacking from a biochemical point of view, this newly developed model incorporates the possibility of a straightforward interpretation within this framework.  2000 Elsevier Science B.V. All rights reserved. Keywords: Predictive microbiology; Primary modelling; Model analysis; Microbial inactivation; Pasteurisation processes; Mild thermal treatment

1. Introduction Due to their high quality level sous vide and cook–chill food products conquered an important *Corresponding author. Tel.: 132-16-32-19-47; fax: 132-1632-19-60. E-mail address: [email protected] (J.F. Van Impe).

part of the food market focussing on convenience. The construction of a global dynamic (i.e., able to deal with time-varying environmental conditions) model for the thermal processing of sous vide or cook–chill food products presents a specific problem: the thermal treatment is less intensive than in canning. The objective is to obtain an optimal trade off between textural and sensorial quality retention on one hand, and reduction of pathogenic and

0168-1605 / 00 / $ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S0168-1605( 00 )00362-7

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spoilage vegetative cells on the other hand. Consequently, surviving thermoresistant micro-organisms (or their spores) may start growing during the subsequent stages of chilling, transport, storage and distribution. This paper is organised as follows. In the next section, an overview of the experimental data used is presented. In Section 3, the design requirements for a non-loglinear, dynamic, and mathematically consistent inactivation model are formulated. Some model types are immediately discarded from further analysis. Sections 4–11 deal with different mathematical modelling approaches available in literature. Within each section, the model considered is shortly described, whereafter results obtained during this research are presented. In Sections 12 and 13, a novel model type is developed and compared with other modelling approaches. In the last section, final conclusions are formulated. Part of the results in this article have been published as Geeraerd et al. (1999).

2. Materials: experimental data used In Table 1, an overview of experimental data used to illustrate model performances is presented. All data were obtained in the framework of EU-project AIR2-CT93-1519.1 A short description of this project can be found in Gibbs (1999). The data of Bacillus cereus and Escherichia coli O157:H7 (VTEC Strains E40705, E30480, E30138 and E30228) originate from the Leatherhead Food Research Association, Leatherhead, UK. Most survivors curves of B. cereus are more or less log-linear, while some survivor curves exhibit a tailing effect. The data of E. coli O157:H7 exhibit both a shoulder and a tail. A representative example is shown in Fig. 1.

1

EU AIR2-CT93-1519 and KU Leuven Research Council Project KUL-COF / 95 / 009: the microbial safety and quality of foods processed by the Sous Vide system of commercial catering.

Table 1 Overview of the experimental data used Micro-organism

Serotype or Strain

Medium

T (8C)

Bacillus cereus spores

F4623

NB (nutrient broth) potatoes in vacuum bags NB (nutrient broth) minced beef in vacuum bags potatoes in vacuum bags TSB (trypticase soy broth)

85, 95 90 85, 90, 95 90 90 60

TSB (trypticase soy broth)

50, 55, 60

minced beef in vacuum bags potatoes in vacuum bags minced beef in vacuum bags 1 effect of added sodium lactate potassium phosphate buffer 1 added beef extract minced beef in vacutainers minced beef in vacuum bags potatoes in vacuum bags minced beef in vacutainers minced beef in vacuum bags potatoes in vacuum bags minced beef in vacutainers minced beef in vacuum bags minced beef homogenate in vacutainers

50, 55, 60 50, 55, 60

F4626

Escherichia coli

O157:H7 (wild) O157:H7 (mutant)

Lactobacillus sake

706

Listeria monocytogenes

4b

Yersinia enterocolitica

O:3 (wild) O:3 (mutant)

55, 60 50, 52, 53, 55 50, 52, 55, 60 50, 55, 60 50, 55, 60 50, 55, 60 50, 55, 60 50, 55 50, 55 50, 55, 60 50, 55 50, 55, 60

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Fig. 1. Typical non-loglinear survivor curve of Escherichia coli O157:H7 processed at 608C.

The only non-pathogenic micro-organism, Lactobacillus sake 706, was studied at our research department in co-operation with Alma University Restaurants, KU Leuven. Obtained results all indicate a sigmoidal shape of the survivor curves. The data of Listeria monocytogenes and Yersinia enterocolitica come from the National Food Centre, Dublin, Ireland. The complete description of the experimental set-up can be found in Doherty et al. (1998), who indicate that, generally, tailing occurred for Y. enterocolitica in potatoes at 508C, and for L. monocytogenes in potatoes at 50 and 558C. In contrast, tailing was not generally present when heated in minced beef.

3. Design requirements Classically, thermal inactivation during sterilisation processes is dynamically described in analogy with chemical kinetics as a first-order decay reaction of the microbial population N (cfu / ml) during the time t (Chick, 1908). dN ]5 2k?N dt

(1)

A first-order differential equation is used because there is a need for a dynamic description when

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dealing with time-varying (and spatially varying) temperatures within the food product. In combination with an Arrhenius-type equation, describing the temperature dependence of the specific inactivation rate k (1 / min), an accurate description of the sterilisation treatment (between 115 and 1308C) focussing on Clostridium botulinum, is feasible. An independent and equivalent description is proposed by Bigelow (1921), introducing the TDT concept (thermal death time) with decimal reduction time D (min), which is the time needed to reduce the bacterial population by one logarithmic unit under isothermal conditions. The temperature dependence of the D value is parameterised through the z value: the temperature increment needed for a 10-fold decrease in D value. By plotting D against temperature, a loglinear relationship is usually obtained indicating a z value which is independent of temperature. Especially in the particular case of C. botulinum, the major public health hazard in low acid canned foods, this TDT concept is widely accepted and applied since decades (see, e.g., Stumbo et al., 1975). For the particular data set at hand about the mild heat inactivation of Bacillus cereus spores (see Table 1), the classical description proved also suitable (results not shown). As indicated higher, a non-loglinear survivor curve, as in Fig. 1, can be observed when a mild heat treatment is applied. For a thorough assessment of this treatment, a very accurate dynamic description of an (eventually) non-loglinear survivor curve is essential. A new dynamic model is needed because the classical first-order inactivation model is structurally not appropriate (see, e.g., Gould, 1989). The following design requirements, presented in two groups, can be formulated in order to describe accurately inactivation curves during a mild heat treatment. The first group is based on experimental observations, as mentioned higher, while the second group will be motivated thereafter. • The model, representing log 10(N) or ln(N) as a function of time, should be able to simulate if necessary a smooth initiation (shoulder) and / or saturation (tail) of the thermal decay. The length of the shoulder, the inactivation rate, and the level of tailing should not be dependent upon each other. This feature implies that the model should encompass a (more or less) loglinear inactivation

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by the selection (after identification on experimental data) of specific parameter values. • The model should be dynamic. Closely related properties are (i) that the model should not depend on N(0) or t 0 explicitly in the right-hand side, and (ii) that for the static equation describing N 5 f(t), the value of N at t 0 , denoted as N(t 0 ), should be equal to the initial condition N(0). Moreover, the model should describe the absolute population (and not the population relative to the initial population), and should be autonomous. These second, mathematically founded model features, are basic requirements for all carefully formulated primary modelling approaches and can be motivated as follows. The dynamic character (i.e., under the form of one or more differential equations or having the possibility to derive a dynamic version by differentiating the static model with respect to time t) is needed in order to be able to describe the influence of, e.g., a temperature increase and decrease within a food product, by integrating the model equation over the time course, taking into account the time dependence of the environmental conditions. Generally, a differential equation has a set of constant parameters (e.g., representing the maximum specific growth rate of the population). The initial condition N(0) and initial time t 0 should not appear explicitly in the right-hand side of the differential equation, because the differential equation should be valid for a whole range of [t 0 ,N(0)] couples. The explicit solution of a dynamic model, using a specific [t 0 ,N(0)] couple, delivers the static version, which is only valid during non-varying environmental conditions (see, e.g., Van Impe et al., 1992; Baranyi et al., 1993). When trying to identify a static model on dynamic experimental data, this can only be done by resetting certain values, e.g., t 0 or N(0), at every environmental change, a procedure which becomes rather involved or even impossible for continuously changing environmental conditions, e.g., an increasing temperature profile during the preheating phase. Observe that by doing so, a discrete time version of the underlying dynamic model is applied. Additionally, for the static equation describing N 5 f(t), the value of N at t 0 , denoted as N(t 0 ), should be equal to the initial condition N(0), because otherwise the initial condition will not be completely

separated of the model parameters and will confound their estimation (France and Thornley, 1984). Moreover, a primary model should describe the absolute population N more precisely, a variance stabilising transformation of it, e.g., ln(N) (Jarvis, 1989), and not the relative population N /N(0), and this for two reasons: (i) the implicit occurrence of N(0) yields problems when this model would be used in subsequent stages of microbial growth and microbial inactivation, and (ii) statistically, by describing the microbial population relatively to the initial population, the assumption is made that the first experimental data point, related to this initial population, is very accurately determined in comparison with all other experimental data points. Considering microbiological practice, this assumption is not acceptable. Finally, the requirement for an autonomous model (i.e., without t in the right-hand side of the differential equation) is the less stringent one. An autonomous model type should be preferred because it can reasonably be assumed that the behaviour of a microbial population does not depend on the absolute time instance. In this research, model types, representing the logarithm of the microbial population as a function of time, were sought in literature, thoroughly analysed and adapted to the specific needs. Different modelling approaches are presented in order of their publication date. If relevant, the following topics are included: (i) description, (ii) adaptations, (iii) dynamic version, (iv) experimental results and / or (v) limitations of the modelling approach. The following model types can immediately be discarded due to their obvious structural limitations. • The model of Cole et al. (1993) is excluded for two reasons: (i) it has a symmetric log-logistic nature, which can not be motivated from an experimental point of view, and (ii) it is based on vitalistic arguments, which will be discussed further in Section 12. • The radiation-based inactivation models of Brynjolfsson (1978) and Porter (1964) can be translated towards heat inactivation kinetics and reformulated in a dynamic version (results not shown). However, they were not developed to incorporate tailing effects and will not further be discussed.

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• The modified Gompertz growth model, introduced by Gibson et al. (1987), is commonly accepted as being a suitable model for describing microbial growth (Zwietering et al., 1990; Garthright, 1991). A dynamic version of this model is developed by Van Impe et al. (1992). Applications of the modified Gompertz growth model to the analysis of inactivation data can be found in Bhaduri et al. (1991) and Linton et al. (1995). The authors indicate the usefulness of this model type for describing sigmoidal survivor curves. This is confirmed in the present research when modelling the static inactivation data of Escherichia coli O157:H7 and Lactobacillus sake (results not shown). However, two modelling problems are inherent to this modified Gompertz type model description: (i) in the static version, N(t 5 0) is not equal to N(0), and (ii) the differential equation does explicitly depend on N(0), which is to be avoided. Therefore, this model type is not further considered.

4. First inactivation model of Casolari

189

combines probability theory with the Maxwellian distribution of energy to obtain the following pseudo mechanistic ( grey box) model: 1 ]]

N(t) 5 N(0) 11B(T )t NA 2 2 2Ed B(T ) 5 ]] exp ]] MH 2 O RT

S D S D

(2)

with N(0) the initial population of micro-organisms, Ed (kcal / mol) the energy necessary for a lethal hit, R (8.314 kcal / mol K) the universal gas constant, T (K) the absolute temperature, MH 2 O (g / mol) the mass of one mole of water molecules and NA (1 / mol) the Avogadro number. A simulation result for this model is presented in Fig. 2 for different inactivation temperatures, with N(0) 5 10 7 cfu / ml and Ed 536 kcal / mol. An increment of the temperature results in a faster decay and a more resistant population requires more energy for a lethal hit (higher Ed ). According to the classification scheme of Cerf (1977), the viewpoint of Casolari is built upon a mechanistic theory of tailing, regarding tailing as a normal feature, bound to the mechanism of inactivation. More will be said about possible explanations for the occurrence of tailing in Section 12.

4.1. Description of the model 4.2. Dynamic equation based on the static version Casolari (1988) attempts to explain the occurrence of the tail in the survivor curve of a population of micro-organisms N (cfu / ml) based on concepts of probability. He assumes that death of the organism is caused by a lethal hit of a water molecule that carries a certain level of energy, higher than the critical level Ed (kcal / mol). As such, Casolari regards tailing as a phenomenon produced by the increasingly low probability of collision between water molecules having more than Ed energy and microbial particles. This viewpoint of Casolari is strengthened by the experiments of Bigelow and Esty (1920), who diluted the initial bacterial concentration and observed an increased thermotolerance, which can be seen as a decrease of the probability of a lethal hit. Observe the similarity with a more general concept known from literature: the water content of a food product has a significant influence on the thermal inactivation rate because the thermal energy, supplied to the food product during the heating process, is transferred mostly via the water fraction (Stumbo, 1973; Tomlins and Ordal, 1976). Casolari

The associated autonomous dynamic model can be calculated to be

Fig. 2. Survivor curve according to the pseudo mechanistic model of Casolari for different temperatures.

A.H. Geeraerd et al. / International Journal of Food Microbiology 59 (2000) 185 – 209

190

dN B(T )(ln N)2 ] 5 2 ]]]] N dt ln N(0) Observe the analogy with Eq. (1) by using the following definition of the decay rate k (1 / min). B(T )(ln N)2 k(T, N, N(0)) 5 ]]]] ln N(0) Instead of the Arrhenius equation, a new nonlinear relation is obtained for the decay rate k (1 / min), depending on the temperature T, the initial and actual microbial population, N(0) and N, respectively.

4.3. Limitations

Fig. 3. Survivor curve according to the second model of Casolari for different temperatures.

The extrapolation of the Maxwellian energy distribution equation, which is theoretically proven for ideal gases, towards microbial inactivation kinetics in liquids leads to a loss of the mechanistic background and the model of Casolari becomes pseudo mechanistic. Although this model describes the tail, occurring in the non-loglinear survivor curve, it still misses the possibility to model the shoulder. Moreover, the explicit occurrence of N(0) in the righthand side of the dynamic version is not acceptable.

5. Second inactivation model of Casolari

5.1. Description of the model As suggested by Casolari (1988), a small adaptation to Eq. (2) is proposed, namely, raise the time to the square. Because of this the model will have more black box or empirical characteristics 1 ]] 11B(T )t 2

N(t) 5 N(0) NA B(T ) 5 ]] MH 2 O

S D S D 2

2 2Ed exp ]] RT

(3)

For the first time, a shoulder can be modelled and a non-loglinear form of the survivor curve can be described (Fig. 3, where N(0) 5 10 7 cfu / ml, and Ed 536 kcal / mol).

5.2. Dynamic equation based on the static version The following autonomous dynamic version is obtained.2 2B(T ) S D 2 ]] (ln N) N ln N(0)

d 2 N (2 1 ln N) dN ]] 5 ]]] ] N ln N dt dt 2

2

2

(4)

The second-order differential equation is a linear function of the population N and of the first derivative of the population dN / dt, but the parameters of it are a nonlinear function of temperature T, N(0) and N. Observe that the initial value of the first derivative is an additional degree of freedom. For positive values of the initial first derivative the shoulder occurs; the more negative the value of the initial first derivative, the faster the inactivation will be.

5.3. Experimental results and discussion Parameter values for experimental data of Y. enterocolitica and E. coli O157:H7 are identified. Inactivation of Y. enterocolitica in minced beef in vacutainers (see Table 1) during the course of three heating processes was recorded. Each heating pro2

In fact, the first derivative can also be written without t in the right-hand side. The equation results in dN / dt 5 0 at t 5 0, which implies a numerical problem when trying to start the integration, and N remains at its initial level N(0).

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cess took place at constant temperature: 50, 55 or 608C. Thirty-two datasets were obtained and each one was modelled with the static model prototype presented in Eq. (3). An example is presented in Fig. 4, left. The only parameter that has to be identified for every dataset is the thermal inactivation energy Ed (kcal / mol). This thermal inactivation energy Ed is revealed to be temperature dependent, which can be seen in Fig. 4, right. Linear regression is possible and results in a descending straight line. This result makes the use of Ed as a parameter independent of

191

temperature, as hypothesised by Casolari (1988), inappropriate. Observe that this regression function only holds within the region where data are available: between 50 and 608C. Outside this area, no assumptions are made concerning the value of the thermal inactivation energy Ed . Data concerning the inactivation of E. coli O157:H7 in TSB (Table 1) are processed in the same way as described above. The results are illustrated in Fig. 5. The thermal inactivation energy Ed exhibits a

Fig. 4. Left: thermal inactivation of Y. enterocolitica at 508C described by the second model of Casolari. One Ed value is identified, associated with this processing temperature. Right: thermal inactivation energy Ed (kcal / mol) for Y. enterocolitica as a function of temperature.

Fig. 5. Left: thermal inactivation of E. coli O157:H7 at 608C described by the second model of Casolari. One Ed value is identified, associated with this processing temperature. Right: thermal inactivation energy Ed (kcal / mol) for E. coli O157:H7 as a function of temperature.

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similar dependence on temperature. Observe the increase of the variance of Ed with increasing temperature, which could be due to problems of reproducibility of experiments encountered at 55 and 608C. However, as the inactivation curve is clearly not described in the most appropriate way, it is difficult to relate properties of the Ed values with the original experimental data.

5.4. Limitations The mathematical properties of Eqs. (3) and (4) have to be evaluated. Because of its probabilitybased mathematics, N always approaches one for increasing time values (or, equivalently, log 10 (N) approaches zero). An important question (which arose partly because it was impossible to obtain a more appropriate description than the one shown in Fig. 5, left), is whether the length of the shoulder is determining the slope beyond. This correlation is proven in Appendix A, demonstrating that the model misses a degree of freedom because of the connection between the slopes of the survivor curve at distinct moments in time: survivor curves exhibiting an initial flat shoulder followed by a relatively fast decay cannot be represented by this model. This problem occurred when trying to describe the inactivation of Lactobacillus sake (results not shown) and occurred also, although to a lesser extent, in Fig. 5. Moreover, the explicit dependence on N(0) in Eq. (4) is not acceptable. As such, several model requirements, as formulated in Section 3, cannot be fulfilled. Results and limitations of the Casolari models have been presented in Herremans et al. (1997).

population. Two types of spores are distinguished: (i) a dormant, viable population ND (cfu / ml), potentially able of producing colonies on / in an appropriate growth medium after activation, and (ii) an active population NA (cfu / ml), able of forming colonies on / in a suitable growth medium. dND ]] 5 2 (k d 1 1 Ka )ND with initial condition ND (0) dt dN ]A 5 k a ND 2 k d 2 NA with initial condition NA (0) dt (5) k d 1 (1 / min) is the first-order inactivation constant of the dormant population ND . Ka (1 / min) is the firstorder activation constant of the dormant spores. k d 2 (1 / min) is the first-order inactivation constant of NA . A model prediction is presented in Fig. 6 (full line), in which log 10 (N) corresponds to log 10 (NA ).

6.2. Inclusion of tailing This model is extended by the introduction of a resistant spore population NR and a sensitive spore population NS , whose summation forms NA . The new model equations are identical to Eq. (5), replacing NA by NS . The resistant population is not inactivated and makes the description of a tail possible, as can be seen in Fig. 6 (dashed line).

6. Inactivation model of Sapru

6.1. Description of the model In a next step, the model of Sapru et al. (1992, 1993) is investigated. These differential equations are inspired on Rodriguez et al. (1988) and Teixeira and Rodriguez (1990). The Sapru model has been derived specifically to describe the activation of microbial spores during sterilisation processes, which implies an initial increase of the activated spore

Fig. 6. Prediction of the model of Sapru et al. (1992) for the inactivation of spores (full line), adapted model (dashed line).

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6.3. Limitations A first limitation is that it is structurally impossible to describe a more or less flat initial shoulder, as the model was designed to describe spore activation and inactivation at high temperatures. Taking into account the design requirements of this research, suppose a slowly increasing shoulder could be acceptable. The time instance t peak at which the shoulder reaches its maximal value (for static conditions) can be calculated analytically.

SS

Ka 1 k d 1 1 t peak 5 ]]]] ln ]]] Ka 1 k d 1 2 k d 2 kd 2

D

S

(Ka 1 k d 1 2 k d 2 )NA (0) 1 Ka ND (0) 2 ln ]]]]]]]]] Ka ND (0)

DD

This highlights the second limitation: the length of the shoulder is completely determined by the slopes before and after t peak . As such, this model cannot fulfil the formulated structural requirements.

7. Inactivation model of Whiting

S

D

5 log 10 (2) 2 log 10 (1 1 exp(b 1 t)] As mentioned in Whiting (1993), this equation yields essentially a straight line, but not completely. In contrast with what could be intuitively expected, this simplification does not coincide with classical linear inactivation kinetics. 2. tl ± 0

S D

S

1 1 exp(2b 1 tl ) N log 10 ]] 5 log 10 ]]]]] N(0) 1 1 exp(b 1 (t 2 tl ))

D

This implies that at t 5 0, N 5 N(0), which is a desirable feature. The model results in a shoulder, followed by a linear decrease. d F1 ± 1 1. tl 5 0

S

2 (1 2 F1 ) 1 ]]]] 1 1 exp(b 2 t)

D

(7)

In this case, two distinct slopes can be distinguished. 2. tl ± 0 This is the most general form, already presented in Eq. (6). After an initial shoulder, two linearly decreasing populations can be observed.

7.2. Inclusion of tailing

S

F1 (1 1 exp(2b 1 tl )) N ]] 5 log 10 ]]]]]] N(0) 1 1 exp(b 1 (t 2 tl )) (1 2 F1 )(1 1 exp(2b 2 tl )) 1 ]]]]]]] 1 1 exp(b 2 (t 2 tl ))

S D

N 2 log 10 ]] 5 log 10 ]]]] N(0) 1 1 exp(b 1 t)

S D

The model of Whiting (1993) is derived from the logistic based model of Kamau et al. (1990) and aims at taking into account a shoulder in the initial part of the survivor curve. Moreover, two populations are distinguished, one of them with larger thermal resistance. This leads to a survivor curve with two distinct regions of linearly decreasing populations, the second with a less negative slope (which is, in this formulation, designated as tailing).

S D

lag period (h). Several special cases can be distinguished. d F1 5 1 1. tl 5 0

2 F1 N log 10 ]] 5 log 10 ]]]] N(0) 1 1 exp(b 1 t)

7.1. Description of the model

log 10

193

D

(6)

with b 1 , maximum specific death rate of major population (1 / h); b 2 , maximum specific death rate of subpopulation (1 / h); F1 , fraction of initial population in major population (2); F2 , 12F1 5fraction of initial population in subpopulation (–); tl , shoulder or

To fulfil the stated model requirements, it is necessary that the tailing exhibits a remaining population. This can be simulated by choosing b 2 5 0.

S D

S

D

F1 (11exp(2b 1 tl )) N log 10 ]] 5log 10 ]]]]]] 112F1 N(0) 11exp(b 1 (t2tl ))

(8) which equals

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194

Fig. 7. Simulation of the inactivation model of Whiting with a constant value for the tail.

S D

N log 10 ]] 5 log 10 N(0)

S

1 1 exp(2b 1 tl ) f (1 2 F1 ) exp(b 1 t) 1 F1 g ]]]]]]]]]]]] 1 1 exp(2b 1 tl ) exp(b 1 t)

Fig. 8. Identification of the inactivation model of Whiting on experimental data of Y. enterocolitica.

kinetics by choosing tl 5 2 ` (although this parameter value is physically unrealistic) and F1 5 1.

D

7.4. Experimental results and discussion A simulation is presented in Fig. 7, where N(0) 5 10 7 cfu / ml, b 1 5 0.16 / min; tl 5 30 min, F1 5 0.99 and b 2 5 0 / min. The length of the shoulder is connected with tl , while the remaining population coincides with 1 3 10 22 of the initial population and is completely determined by F1 . The model has an inflection point, which can be calculated to be

S

7.5. Limitation

D

1 1 F1 exp(2b 1 tl ) 1 t infl 5 ]] ln ]]]]]] 1 tl 2 b1 1 2 F1

7.3. Dynamic equation based on the static version A first derivative is sufficient to eliminate the time in the right-hand side of the equation.

S

D

N(0)(1 2 F1 ) dN ] 5 2 b 1 1 2 ]]]] dt N N 2 N(0)(1 2 F1 ) ? 1 2 ]]]]]]] N(0)F1 (1 1 exp(2b 1 tl ))

F

G

The model is applied to datasets of Y. enterocolitica. An example can be found in Fig. 8, describing the thermal inactivation at 508C in minced beef in vacutainers. Observe the adequate description of the inactivation curve.

?N

(9)

The model encompasses classical linear inactivation

This four parameter inactivation model is able to cope almost completely with the formulated model requirements: (i) a dynamic version can be obtained, and (ii) the model can describe shoulders and tails in the inactivation curve in an independent way. However, the explicit occurrence of (i) N(0) and (ii) F1 , which is related with N(0), in the right-hand side of Eq. (9), is not acceptable and limits the applicability of the model to static conditions. A minor remark is that tl does not visually coincide with a time period where the bacterial population remains at the inoculum level (which was formulated as such in Whiting (1993)). This can be seen in Fig. 7.

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8. Growth model of Baranyi

8.1. Description of the model The growth model of Baranyi and Roberts (1994), which has become the standard growth model in many research groups working in the field of predictive microbiology, can be summarised as follows. Suppose an intracellular substance P(t) (units / cell), which must accumulate in a certain amount before growth can start, grows exponentially and controls the specific growth rate according to Michaelis– Menten kinetics. On the other hand, the transition of the exponential growth of the microbial cells x(t) (cfu / ml) to the stationary phase is described by a logistic type limiting function. With the notation P(t ) q(t) 5 ] K P (–) (where KP (units / cell) stands for the Michaelis–Menten constant), this can be summarised as: dq ] 5 mmax q(t) dt dx q(t) ] 5 mmax ]]] dt q(t) 1 1

S

DS

D

x 1 2 ]] ? x x max

(10)

The model has four degrees of freedom: two initial states (q(0) and x(0)) and two parameters ( mmax and x max ).

8.2. Description of inactivation curves Baranyi et al. (1996) propose to consider an inactivation curve as the mirror image of a growth curve. The following procedure is suggested. 1. hi is the natural logarithm of the measured cell concentrations of an inactivation curve (i 5 0, . . . ,k 2 1 with k the number of experimental points). Choose a fixed value hfix which is greater than any hi . Redefine hi as hfix 2 hi at each time instant. The new hi values represent a growth curve being the mirror image of the original inactivation curve. 2. This growth curve can be modelled with Eq. (10). As a result, 2 mmax represents the maximum specific inactivation rate, the lag phase corresponds to the shoulder, and the stationary level coincides with the tail.

195

As demonstrated in Baranyi et al. (1996), this dynamic inactivation model can describe shoulders and / or tails as well as the possible loglinear decrease of a microbial population in an adequate way. All formulated structural model requirements are fulfilled. However, the authors prefer a classical log-linear kinetic in order to obtain a fail-safe estimate of the inactivation rate. It should be noted that as such, an accurate description of the non-loglinear survivor curve is not possible. Moreover, the approach also neglects tailing, if occurring, and can therefore not be designated as fail-safe.

9. Inactivation model of Chiruta Chiruta et al. (1997) synthesised a new empirical and non-loglinear model from data for three vegetative bacteria: Pseudomonas fluorescens 172, Listeria monocytogenes SLCC 5764, and Escherichia coli ATCC 25922.

9.1. Description of the model

F

ln 1 2 log 10

S DG N ]] N(0)

5 e 1 v ? ln(t) 1 V ? (ln(t))2

with e, v and V [–] critical coefficients. This means, when comparing with ln(N /N(0)) 5 2 k t (which is the static solution of Eq. (1)), 2.303 k 5 ]] ? [exp (e 1 v ? ln(t) 1 V ? (ln(t))2 ) 2 1] t (11) Observe the time-dependent rate constant for inactivation, as postulated by Chiruta et al. (1997).

9.2. Dynamic equation based on the static version The dynamic version reads as follows. dN ] 5 2k N dt dk 2.303 ] 5 ]] s1 2 (1 2 v 2 2 ? V ? ln(t)) ? exp [e 1 v ? ln(t) 1 V ? (ln(t))2]d dt t2

Due to the complex influence of time t on the inactivation rate k (Eq. (11)), t cannot be eliminated from the right-hand side of the equation.

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9.3. Analysis of the model

10.2. Dynamic equation based on the static version

For the structural analysis of this inactivation model, which highlights the artificial character of it, reference is made to Appendix B.

It is necessary to twice derivatise Eq. (12) to eliminate the time from the right-hand side of the equation.

S D

S D

9.4. Limitations Chiruta et al. (1997) already mention that their model is purely empirical displaying a high degree of fit for their particular case studies. The analytical results of Appendix B indicate the following disadvantages. 1. The initial population always approaches 2 `. Depending on the parameter values, after a very short time period, ln[N(0)] is reached / nearly reached again. 2. An initial raise, followed by a decrease in the population, is possible, but before t 5 1 s the population equals (or nearly equals, depending on the parameter values) the initial population again. 3. For t larger than 1 s, the population displays a non-loglinear decrease reaching 2 ` for t → 1 `. As such, model requirements cannot be fulfilled.

d 2N dN 2 1 dN N ]] ] ] 2 2 l ? ] 2 l 2 ? log 10 ]] 5 2 dt N dt N(0) dt ? ln(10) ? N

This model can be reformulated as two coupled first-order differential equations dN ] 5 2k?N dt dk N ] 5 2 2lk 1 l 2 ln(10)log 10 ]] dt N(0)

S D

As such, the classical inactivation kinetics can be recognised (Eq. (1)). Observe that the model uses two states (N and k) and one additional parameter ( l).

10.3. Analysis of the model At t 5 0, log 10 (N) equals log 10 (N(0)), which is a desirable feature. Fig. 9 displays the satisfying curvature of the model at limited time instances for the parameter values, presented in Daugthry et al. (1997) for S. aureus 196E at 608C in skimmed milk, namely

10. Inactivation model of Daughtry

10.1. Description of the model Daugthry et al. (1997) developed an exponentially damped polynomial model. It is indicated that the overall fit of the model is significantly better than the log-linear model for the data sets considered (inactivation of Escherichia coli and Staphylococcus aureus as function of temperature and sugar concentration). The model reads as follows.

S D

N log 10 ]] 5 2 k d ? t ? exp(2l ? t) N(0)

(12)

with k d (1 / h) the initial inactivation rate, and l (1 / h) the damping coefficient.

Fig. 9. Simulation of the Daughtry model.

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remains constant, followed by a loglinear decrease. Moreover, the total microbial population N is supposed to be divided over two subpopulations N1 and N2 (as in the model of Sapru et al. (1992), discussed in Section 6), differing in heat inactivation constant (k 1 . k 2 ). The model reads as follows. N1 (t) dN ]1 dt N2 (t) dN ]2 dt

5 5 Fig. 10. Simulation of the Daughtry model for increasing time values.

l 5 2.322 3 10 22 / min and kd 5 0.3456 / min. However, the model behaviour at increasing time instances is not satisfactory, as can be seen in Fig. 10. The lowest point of the curve, reached at t min , can be calculated to be

5 N01

(N01 $ 0;t 0 # t # t lag )

5 2 k 1 N1 (t $ t lag ) 5 N02

(N02 $ 0;t 0 # t # t lag )

5 2 k 2 N2 (t $ t lag )

with t 0 the initial time, N(t) 5 N1 (t) 1 N2 (t), and N(0) 5 N01 1 N02 . The model was evaluated using 20 survival curves of Staphylococcus aureus originating from Whiting et al. (1996). During a treatment under constant environmental conditions, and using the notation f 5 N01 /(N01 1 N02 ), the above model reduces to for (t 0 # t # t lag ):

S D

log 10

N ] N(0)

5

5

0 for (t $ t lag ): log 10 ( f exp(2k 1 (t 2 t lag )) 1 (1 2 f ) exp[2k 2 (t 2 t lag )])

t min 5 1 /l With the given parameter value for l, t equals 43 min at that moment. At t → `, N equals N(0) again.

10.4. Limitations Although the model can be useful for limited time periods, its behaviour with increasing time t is unacceptable: tailing cannot be described adequately. A shoulder cannot be included in this model. Moreover, the appearance of N(0) in the right-hand side of the dynamic model is not allowed. As such, the stated model requirements cannot be fulfilled.

11. Inactivation model of Xiong

(13)

(14)

11.2. Static character of the model As formulated in the design requirements, the inactivation model should be dynamic in order to be able to deal with time varying temperature profiles. The model of Xiong is not completely dynamic, because it is not differentiable at t 5 t lag . This implies that a sudden temperature change, e.g., a step profile, can only be dealt with by resetting the initial time t 0 at the instant of the temperature change and again using the static model equations from that moment on (as explained in Section 3). This resetting of a model variable is only possible for constant temperature profiles with some occasional discontinuities (step functions).

11.1. Description of the model 11.3. Limitation The model of Xiong et al. (1999) is based on the Buchanan et al. (1993) inactivation model (which, on itself, is extended in Buchanan et al. (1997) and ´ Breand (1998)) and assumes a shoulder region (indicated by t lag ) during which the population level

If realistic time-varying temperature profiles observed in the practice of food preservation occur (e.g., more or less linear increase or decrease), the model of Xiong et al. can only be used if the

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temperature profile is approximated by a (finite) number of temperature steps, a procedure which could become numerically involved. Even the socalled dynamic version, presented in Eqs. (13) is not usable, because of the separate model equations valid before and after t lag . As a conclusion, it can be stated that the model of Xiong et al., like all models which are (partially) static or discontinuous (e.g., Hayakawa (1982) or Buchanan et al. (1993)), is unable to deal with realistic temperature fluctuations in a consistent way. In other words, the model cannot deal with the previous history of the microbial population.

critical molecules, needs to be inactivated (single-hit multiple-target phenomenon). In such a case, the type of damage is cumulative rather than instantly lethal. As a direct consequence of hypotheses 1 and 3, it can be stated that the cells are protected to the heat by components outside the cell (namely, other cells or specific molecules in the medium, respectively). On the other hand, hypotheses 2 and 4 state that a certain component within the cell or a number of the same or different components, is critical for its survival. In both instances, assume a certain component C (units / cell) undergoing heat inactivation following a first-order relationship.

12. A novel inactivation model

dC ] 5 2 k max ? C dt

12.1. Derivation of the model 12.1.1. Modelling the shoulder of the inactivation curve According to Mossel et al. (1995), pp. 84, 85, and 94, three explanations are possible for the occurrence of a shoulder in an inactivation curve. 1. Organisms being present in groups or clumps, each clump is represented, upon culturing in a solid medium, by one colony. The length of the shoulder coincides with the time before all but one organism in such a clump have been killed. This is a particular case of a single-hit multipletarget phenomenon (Cerf, 1977). 2. Alternatively or concurrently, the shoulder represents a period during which the cells are able to resynthesise a vital (i.e., critical) component and death ensues only when the rate of destruction exceeds the rate of synthesis. 3. Moreover, proteins or fats in the medium could lead to an increased heat resistance. Proteins could reduce the loss of solutes, stabilise the membrane, or have a buffering effect on low pH values, while fat particles could cause an indirect reduction of water activity due to the increasing solubility of water in fat at raising temperatures. A fourth explanation, related both with the first and the second ones, is cited by Moats et al. (1971) and Adams and Moss (1995): a large number of the same critical molecules, or a large number of different

With respect to hypothesis 2, this relationship can be regarded as a simplification of Eq. (11) in Reichart (1994), where the inactivation of a critical structure in the microbial cell (DNA, enzyme, membrane, . . . ) is presented as function of (i) the concentration of the critical structure, (ii) the concentration of the molecule responsible for the lethal effect, and (iii) a reaction constant, depending on temperature according to Eyring’s theory. The research of Reichart deals with the inactivation of Escherichia coli, which is presumed to be dependent on the inactivation of the critical structure. In the beginning of the inactivation, a whole pool of components is presented around (hypotheses 1 and 3) or in each cell (hypotheses 2 and 4). Gradually, this pool is destroyed. As such, the shoulder is obtained by introducing the following Michaelis–Menten kinetics based adjustment function a in the specific inactivation rate of the population itself. The formulation of a is chosen in order to obtain a value of approximately zero at the beginning of the inactivation, and a value of approximately 1 towards the end of the shoulder region. Cc C 1 a 5 1 2 ]] 5 1 2 ]] 5 ]] C 1 Kc 1 1 Cc 1 1 Cc where Kc (units / cell) stands for the value of C where a equals half of its final value 1. Cc [–] is defined as C /Kc , which can be interpreted as a measure of the physiological state of the population.

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12.1.2. Modelling the linear part and the tailing of the inactivation curve The loglinear part of the inactivation curve is modelled with first-order kinetics (k max ). The tailing effect is reflected in the additional factor s1 2 Nres /Nd which implies the existence of a subpopulation Nres . Observe that this factor is inspired by, but not equivalent to the classical logistic formulation (see, e.g., France and Thornley, 1984). dN ] 5 2k?N dt where

k

S

D

Nres 5 k max ? a ? 1 2 ] N

(15)

The motivation for this subpopulation Nres can be found in Table 2, but, at this moment, no interpretation is available for the selection of a logistic-type inspired tailing model. Nevertheless, Table 2 illustrates how the use of Nres can be interpreted from the different fundamental viewpoints highlighted by Cerf (1977). The tailing occurs as a consequence of the more resistant subpopulation Nres (Theories 1 and 2a) or as a consequence of experimental artefacts (Theory 2b). Cerf concludes in his review on tailing that at that moment no decision could be made between the vitalistic and the mechanistic theories. More recently, Casolari (1988) indicates two types of experiments which should be performed in order to determine whether the vitalistic theory (permanent difference in resistance, Theory 1) or the adaptation theory (acquired difference in resistance, Theory 2a.2) could be a valuable explanation. These theories

199

have in common that a certain distribution of heat resistance in a population is assumed. 1. If one of these two theories would be correct, surviving micro-organisms should have a higher degree of resistance than the majority of the parent population. However, although these type of experiments have been conducted, all authors failed to show greater resistance (see references in Casolari, 1988). 2. The distribution of heat resistance also implies that survivor curves would become more loglinear if decreasingly smaller fractions of the population would be subjected to the heat treatment. However, an extended range of experimental data (see references in Casolari, 1988) gave evidence for an increasing thermal resistance for decreasing initial number of micro-organisms. Casolari builds his model based on a mechanistic theory of tailing, regarding tailing as a consequence of the mechanism of inactivation (Theory 2a.1): a lethal hit by a water molecule (see Section 4). These results also indicate that the approach of Cole et al. (1993), where a vitalistic logistic based inactivation model is developed, is not substantiated by the extended research of Casolari. Observe that the elucidation of the tailing phenomenon from a biochemical, chemical or physical viewpoint is beyond the scope of this research, focussing on the modelling aspects.

Table 2 Motivation for modelling tailing as a subpopulation Nres (third column) within the frame presented by Cerf (1977) (first and second columns) Theory

Subdivision: tailing is . . .

Nres equals a subpopulation which is . . .

1. Vitalistic

. . . a normal feature

. . . very resistant to the heat

2a. Mechanistic

. . . a normal feature, bound to: 1. the mechanism of inactivation 2. the mechanism of resistance

. . . unaccessible for the heat . . . adapted to the heat

2b. Mechanistic

. . . an artefact, due to: 1. genetic heterogeneity 2. heterogeneity of treatment 3. clumping (single-hit, multiple-targets) 4. enumeration of survivors

. . . genetically more resistant . . . not receiving the same lethal dose . . . not completely inactivated . . . very low in number, implying high variability

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12.1.3. Complete model description The complete model with two parameters (k max and Nres ) and two states (N and Cc ) can be described as

where

dN ] 5 2k?N dt dCc . ]] 5 2 k max ? Cc de Nres 1 k 5 k max ? ]] ? 1 2 ] 1 1 Cc N

S

DS

(16)

D

This model has four degrees of freedom: two parameters (k max and Nres ) and two initial states (N(0) and Cc (0)). The model encompasses loglinear inactivation by the selection (after identification on experimental data) of a very low value for Cc (0) and Nres , implying the absence of a shoulder and a tail, respectively. Remark the fact that the value of k max influences the length of the shoulder through the inactivation of the protecting (hypotheses 1 or 3) or critical (hypotheses 2 or 4) components. As such, the influence of the initial physiological state of the cells (Cc (0)) can be separated from the influence of the environment (k max ). According to this model description, for cells in the same initial physiological state, having experienced exactly the same prehistory, the shoulder will be inversely proportional to the inactivation rate. For static conditions, the model reduces to

Fig. 11. Inactivation curve of Lactobacillus sake at 558C described by the newly developed inactivation model.

N(t) 5 (N(0) 2 Nres ) ? exps 2 k max ? td

S

D

1 1 Cc (0) ? ]]]]]]] 1 Nres 1 1 Cc (0) exp(2k max ? t)

(17)

In this expression, N(t 0 ) equals N(0), which is a desirable feature. Observe that, more generally, it could be assumed that the inactivation rate k max in Eqs. (16) or (17) is not the same for Cc as for N. This would imply an extra degree of freedom which is not necessary at this moment, given the nonloglinear behaviour of the experimental survival curves reported thus far.

12.2. Experimental results and discussion Figs. 11 and 12 represent two typical inactivation curves adequately modelled by Eq. (16). The model

Fig. 12. Inactivation curve of Listeria monocytogenes in minced beef in vacutainers at 508C described by the newly developed inactivation model.

is dynamic and fulfils all formulated model requirements. Further experimental evaluation of this model is the subject of ongoing research. The design of optimal experiments for the temperature dependence of k max , using the Arrhenius type model, is presented in Versyck et al. (1999).

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13. Comparison between the new model and other modelling approaches In this section, the newly developed inactivation model is compared with two other modelling approaches discussed in previous sections: • the Whiting type model, because it is useful for a single inactivation step. However, it should be kept in mind that the explicit occurrence of N(0) in its dynamic version is an inherent limitation of this modelling approach making it not suitable for subsequent growth / inactivation steps. • the Baranyi growth model, based on its ability to fulfil the formulated model requirements when applied to the mirror image of a survival curve and its actual widespread use as growth model in predictive microbiology groups around the world.

13.1. Comparison with the Whiting type model The dynamic version of the Whiting model (1993) in case of horizontal tailing has been calculated in Eq. (9). Comparing this expression with the new model presented in Eq. (16) reveals that these two models are reparameterisations of the same expression: the transformations needed are derived in Appendix C and results are summarised in Table 3. It has to be remarked that the transformation between Cc (0), k max and tl implies that the shoulder length (which is related with tl ) is inversely proportional to the inactivation rate for cells in the same initial physiological state, as stated in Section 12. It is striking how a rather empirical approach (the Whiting type model) leads to the same expression as the newly developed model. However, if these model types are to be used in a

201

global process model, describing subsequent stages of possible inactivation and / or growth, the new model type is to be preferred because of the clear separation between initial state variables and model parameters. Observe that these results, together with previous results, imply the necessity to use two state equations for modelling microbial inactivation: one for the microbial population itself (e.g., in cfu / ml), and one for indicating the physiological state of the population, because the behaviour of the same initial population N(0) can be highly dependent on the previous history (see, e.g., Jørgensen et al., 1998).

13.2. Comparison with the Baranyi model The Baranyi growth model when applied to the mirror image of an inactivation curve, according to the procedure described in Baranyi et al. (1996) and summarised in Section 8, reads as follows dq ] 5 mmax q(t) dt d(x fix /x) q(t) ]]] 5 mmax ]]] dt q(t) 1 1

S

DS

D

x fix /x 1 2 ]] ? x fix /x x max (18)

Herein, x fix stands for the fixed value hfix expressed in absolute cell concentrations. When comparing with the newly developed model (16), it can be shown that the two models can be reparameterised towards each other. Transformations are presented in Table 4. As such, both models describe exactly the same survival curve when identified on experimental data points. The transformation between q(0) and Cc (0) im-

Table 3 Analytical equivalence between the Whiting type inactivation model and the new inactivation model

Table 4 Analytical equivalence between the Baranyi growth model, applied to the mirror image of an inactivation curve, and the new inactivation model

Whiting type model degrees of freedom

New model degrees of freedom

Transformations

Baranyi model degrees of freedom

New model degrees of freedom

Transformations

N(0) b1 tl F1

N(0) k max Cc (0) Nres

– k max 5 b 1 Cc (0) 5 exp(k max ? tl ) Nres 5 N(0)(1 2 F1 )

N(0) mmax q(0) x max

N(0) k max Cc (0) Nres

– k max 5 mmax Cc (0) 5 1 /q(0) Nres 5 x fix /x max

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plies that cells at the end of the lag phase (high q value) are more susceptible to an inactivation treatment than cells in early lag phase (low q value). This phenomenon is known in literature (Jay, 1992, p. 340; or Mossel et al., 1995, p. 86). Moreover, the combination with a transformation derived for the comparison with the Whiting type model, namely, Cc (0) 5 exp(k max ? tl ), also implies the following equality

14. Conclusions

exp(k max ? tl ) 5 exp( l ? mmax ) 2 1

• The two models, representing log10(N) or ln(N) as a function of time, are able to simulate independently a smooth initiation (shoulder) and / or saturation (tail) of the thermal decay. The models encompass loglinear inactivation by the selection (after identification on experimental data) of specific parameter values. • The two models are formulated as coupled autonomous differential equations, independent of N(0) or t 0 . For the static versions, the value of N at t 0 , denoted as N(t 0 ), is equal to the initial condition N(0).

with l the lag phase as defined in Baranyi and Roberts (1994). As such, the value of l is slightly larger than the value of tl . However, the transformation between q(0) and Cc (0) does not imply that the components P for growth or C for inactivation would be identical. The Baranyi inactivation model can not incorporate specific biochemical knowledge about the shoulder and tailing phenomenon as it was specifically designed to describe the growth of micro-organisms. The new model of Section 12 is to be preferred because of the possibility to be interpreted or, if necessary, adapted from a mechanistic point of view if, e.g., the critical component(s) and / or the mechanisms of their inactivation would be identified, or, if the tailing phenomenon would be elucidated. Additionally, taking the mirror image of experimental data points, by using Eq. (18), remains a rather artificial way for describing them.

The extensive analytical analysis of a whole range of inactivation models within the framework of the formulated structural model design requirements necessary to describe accurately microbial inactivation during a mild heat treatment, is summarised in Table 5. The most important result is the selection of two models fulfilling these model requirements.

From a strictly mathematical point of view, it is illustrated that the Baranyi growth model applied to the mirror image of a survival curve (Section 8) is a reparameterisation of the new model proposed in Section 12. Also, the Whiting type model and the new model are reparameterisations of the same expression,

Table 5 Summary of the analytical results Model type

Shoulder

a

Tail

Limitations

possible

autonomous

no N(0)

1 1

1 1

2 2

Casolari I (1988) Casolari II (1988)

2 1

Sapru et al. (1992)

1b

1

1

1

1

Whiting (1993) Baranyi et al. (1996) Chiruta et al. (1997)

1 1 2

1 1 2

1 1 1

1 1 2

2 1 1

Daugthry et al. (1997)

2

2

1

1

2

Xiong et al. (1999) New model

1 1

1 1

2 1

n.a.a 1

n.a. 1

a b

1 1

a

Dynamic version

Not independent slopes at different time instances Not independent length of the shoulder No interpretation possible Unrealistic at small time instances Unrealistic at large time instances

1, the model can fulfil this model requirement; 2, the model cannot fulfil this model requirement; n.a., not applicable. If a slowly increasing shoulder could be acceptable.

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causing the model descriptions to be completely coinciding when identified on static experimental data. However, the Whiting model can not cope with subsequent stages of growth and inactivation. Only the new model offers the possibility to be interpreted or, if necessary, adapted from a mechanistic point of view if, e.g., the critical component(s) and / or the mechanisms of their inactivation would be identified, or, if the tailing phenomenon would be elucidated. As such, it is to be preferred.

Acknowledgements This research has been supported by the Research Council of the Katholieke Universiteit Leuven as part of projects COF / 95 / 009, COF / 98 / 008, OT / 99 / 24 and PDM / 99 / 096, the Belgian Fund for Scientific Research (FWO-Flanders) as part of project G.0267.99, the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, the Belgian Ministry of Small Enterprises, Traders and Agriculture as part of project S-5856 and the European Commission as part of projects AIR2 CT93-1519, and FAIR CT97-3129. Dr. Elena ´ Gonzalez-Fandos is kindly acknowledged for generating the experimental data of Lactobacillus sake. Dr. T.A. Roberts, Food Safety Consultant, Reading (UK) is gratefully acknowledged for several helpful suggestions. The scientific responsibility is assumed by its authors.

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function of w and the inactivation energy Ed through the parameter B. 1 ]]

(1 1 Bt 21 )2 N(0) 11w 2 Bt 21 b 5 w ]]]]]]] 1 ]] (1 1 w 2 Bt 12 )2 N(0) 11Bt 21

1

2

a

A simulation result of the relationship between a and b is presented in Fig. 13, where t 2 5 2 ? t 1 5 20 min, N(0) 5 10 9 cfu / ml, and T 5 323 K. By varying Ed , different combinations of a and b can be obtained. The slope at a certain time instant t 1 (a ) is determined by the value of Ed at that moment, as can be seen from Eq. (19). Observe that in most of the cases, two Ed values can be found resulting in the same slope (e.g., an Ed of 36 kcal / mol results in the same a as an Ed of (approximately) 36.37305 kcal / mol for the parameter values of Fig. 13). As such, the slope at t 2 ( b ) can only take on two distinct values, i.e., the two slopes are completely connected through Ed . For Ed →` or 2 `, it can be shown that a and b equal zero. For the selected parameter values, the minimum value for a equals 2 7.011 ? 10 7 cfu /(ml min) and the minimum value for b 2 3.506 ? 10 7 cfu /(ml min). It can be concluded that the slopes of the survivor curve at distinct moments in time are strongly correlated: survivor curves with a close to zero and

Appendix A. Structural limitation of the second model of Casolari The slope at every instant, e.g., at an (arbitrary) time instant t 5 t 1 and at another time instant t 2 (t 2 5 w ? t 1 with w . 1), can be specified by the first derivative of Eq. (3).

* dN b 5 ]* dt dN a 5] dt

1 2 2Bt 1 ln N(0) ]] 5 ]]]]] N(0) 11Bt 21 2 2 t1 (1 1 Bt 1 ) 1 2 2wBt 1 ln N(0) ]] 5 ]]]]] N(0) 11w 2 Bt 12 2 2 2 t2 (1 1 w Bt 1 )

(19)

The combination of these equations results in a nonlinear relationship between a and b, which is

Fig. 13. Illustration of the inherent structural limitation of the second model of Casolari: the slope b on one time instance is correlated with the slope a on a preceding time instance.

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b strongly negative (i.e., inactivation preceded by a shoulder) cannot be represented by this model.

are (i) t 5 1 ` and (ii) the solutions of exp[e 1 v ? ln(t) 1 V ? (ln(t))2 ] 5 1. The second option means

e 1 v ? ln(t) 1 V ? (ln(t))2 5 0 Appendix B. Structural analysis of the model of Chiruta Out of Eq. (11), the derivative of k with respect to the time t can be calculated to be dk 2.303 ] 5 ]] s1 2 (1 2 v 2 2 ? V ? ln(t)) dt t2 2

? exp [e 1 v ? ln(t) 1 V ? (ln(t)) ]d The time instances t where dk / dt 5 0 can only be found numerically and are the solutions of (1 2 v 2 2 ? V ? ln(t))? exp [e 1 v ? ln(t) 1 V ? (ln(t))2 ] 51

(20)

Moreover, dk / dt 5 0 is also fulfilled at t 5 1 `. All subsequent simulations will be performed for ln(N) as a function of time: ln(N) 5 ln(N0 ) 2 k t

(21)

For t larger than 1 and assuming e, v and V are all positive (no information concerning e, v and V was provided in Chiruta et al. (1997)), k is a positive number (Eq. (11)), indicating that ln(N) will be smaller that ln(N0 ). For k 5 0 / s, ln(N) equals ln(N0 ) and for a negative k value the actual population ln(N) will be larger than ln(N0 ). The time instances where k 5 0 / s (out of Eq. (11))

The equation can only have solutions for t smaller than 1 s (otherwise every term is positive). The solutions are ]]]] 2 v 6Œv 2 2 4 ? V ? e t 5 exp ]]]]]]] (22) 2V

S

D

Two possibilities arise. v 2 2 4 ? V ? e . 0 The following parameter values are used for simulation of the model: e 5 0.005, v 5 0.2 and V 5 0.01. In this case, k 5 0 / s at t 5 2.11 ? 10 29 s and t 5 0.9753 s (Eq. (22)). Fig. 14, left, displays the very first part of the model when N0 is defined as 10 8 cfu / ml. ln(N0 ) is indicated with ‘o’. It can be calculated that ln(N) approaches 2 ` at t 5 0 s (not displayed for convenience). Immediately after this starting point, ln(N) is smaller than ln(N0 ) because k is larger than 0 / s (Fig. 14, right, and Eq. (21)). Remark that k at t 5 0 s approaches `. Furthermore, the increasing t is outweighed by the decreasing k-value. As a result, the product k ? t in Eq. (21) becomes smaller and ln(N) raises. At t 5 2.11 3 10 29 s, k equals 0 and ln(N) equals ln(N0 ). Simulation results indicate that dk / dt 5 0 between 5310 29 and 6310 29 s (which is also a solution of Eq. (20)). In Fig. 15, left, the situation beyond this point is

Fig. 14. Simulation of the Chiruta model for t up to 1 3 10 27 s.

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205

Fig. 15. Simulation of the Chiruta model for t up to 8 3 10 23 s.

shown: ln(N) is higher than ln(N0 ) because k , 0 / s. Observe the different scale on the y-axis in Fig. 15, right when compared with Fig. 14, right. The increasing k value, becoming lesser and lesser negative is outweighed by the increasing t, causing the product k ? t to become more and more negative. 25 Around t 5 5 3 10 s, the product k ? t reaches its minimum (negative) value. As a result, ln(N) attains its maximum. Afterwards, the product k ? t raises (but is still negative) and ln(N) lowers. At t 5 0.9753 s, ln(N) equals ln(N0 ) again (Fig. 16, left). For larger t-values, k is positive (Fig. 16, right) and ln(N) is further decreased. Simulation results (Fig. 17) indicate that dk / dt 5 0 between 3.22

and 3.23 s (which is also a solution of Eq. (20)). The decrease only stops for t → `, where dk / dt 5 k 5 0 and ln(N) approaches 2 `. v 2 2 4 ? V ? e , 0 No real solutions can be found for t (Eq. (22)), which implies that k is always positive. The situation is simulated for N0 5 10 8 cfu / ml, e 50.001, v 5 0.004 and V 5 0.05 in Fig. 18. Again, ln(N) approaches 2 ` at t 5 0 s. Hereafter, ln(N) is smaller than ln(N0 ) because k is larger than 0. ln(N) raises because of the fact that the increasing t is outweighed by the decreasing k value. Simulation results indicate that dk / dt 5 0 between 0.95 and 0.96 s (which is also a solution of Eq. (20)). At this point, k is only slightly larger than zero, and

Fig. 16. Simulation of the Chiruta model for t up to 1.5 s.

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Fig. 17. Complete simulation of the Chiruta model.

Fig. 18. Simulation of the Chiruta model for k . 0.

ln(N) is only slightly smaller than ln(N0 ). Afterwards, the population decreases again. A second point where dk / dt 5 0 can be found between 9 and 9.2 s. The increasing t outweighs the decreasing value of k and the population continues to decrease.

Appendix C. Structural comparison of the Whiting type inactivation model and the new model According to Eq. (9), the dynamic version of the Whiting type model reads as follows

F

dNwhi Nwhi 2 N0 (1 2 F1 ) ]] 5 2 b 1 ? 1 2 ]]]]]] dt N0 F1 (1 1 exp(2b 1 tl ))

S

G

D

N0 (1 2 F1 ) ? 1 2 ]]] ? Nwhi Nwhi The comparison of this expression with the new model (Eq. (16)), dC ]c 5 2 k max ? Cc dt dNnew Nres 1 ]] 5 2 k max ? ]] ? 1 2 ]] ? Nnew dt 1 1 Cc Nnew

S

DS

D

A.H. Geeraerd et al. / International Journal of Food Microbiology 59 (2000) 185 – 209

leads to k max 5 b 1 Nres 5 N0 (1 2 F1 )

(23)

The following equality needs to be proven before it can be stated that the two models are reparameterisations of the same expression. Nwhi 2 N0 (1 2 F1 ) 1 1 2 ]]]]]]0]] N0 F1 (1 1 exp(2b 1 tl )) 1 1 Cc

207

dCc Nwhi 2 Nres 5 ] ]]] dt Cc (1 1 Cc )

Using the first equation of the new model, the equality becomes dNwhi Nwhi 2 Nres ]] 5 2 k max Cc ]]] dt Cc (1 1 Cc ) which can be reformulated towards

Using the equality N0 F1 5 N0 2 Nres (Eq. (23)), this can be rewritten as (N0 2 Nres )(1 1 exp(2b 1 tl )) 2 Nwhi 1 Nres 1 ]]]]]]]]]]]] 5 ]] 1 1 Cc (N0 2 Nres )(1 1 exp(2b 1 tl )) Cc (N0 2 Nres )(1 1 exp(2b 1 tl )) 2 (Nwhi 2 Nres )(1 1 Cc ) 50 Cc (N0 2 Nres )(1 1 exp(2b 1 tl )) 1 Nres (1 1 Cc )

S

DS

D

dNwhi Nres 1 ]] 5 2 k max ? ]] ? 1 2 ]] ? Nwhi dt 1 1 Cc Nwhi This is equal to the second equation of the new model, and, as such, completes the proof. The transformation between Cc (0), b 1 and tl can be obtained by evaluating Eq. (25) at t 5 0 Cc (0) 5 exp(b 1 ? tl )

5 Nwhi (1 1 Cc ) References

which leads to Cc (N0 2 Nres )(1 1 exp(2b 1 tl )) 1 Nres (1 1 Cc ) Nwhi 5 ]]]]]]]]]]]]] 1 1 Cc (24) Differentiating this equation with respect to time t reads as follows dNwhi ]5 dt dCc dCc ] f (N0 2 Nres )(1 1 exp(2b 1 tl )) 1 Nres g (1 1 Cc ) 2 ] [Nwhi (1 1 Cc )] dt dt ]]]]]]]]]]]]]] (1 1 Cc )2

Substituting the following equality (Eq. (24)) (Nwhi 2 Nres )(1 1 Cc ). 1 1 exp(2b 1 tl ) 5 ]]]]]] (N0 2 Nres )Cc leads to

F

G

dC (Nwhi 2 Nres )(1 1 Cc ) dCc ]c ]]]]] 1 Nres 2 ] Nwhi dNwhi dt Cc dt ]] 5 ]]]]]]]]]]] dt (1 1 Cc )

F

G

F

G

dC Nwhi (1 1 Cc ) 2 Nres dCc ]c ]]]]] 2 ] Nwhi dt Cc dt 5 ]]]]]]]]] (1 1 Cc ) dC Nwhi Nres dCc ]c ] 1 Nwhi 2 ] 2 ] Nwhi dt Cc Cc dt 5 ]]]]]]]]] (1 1 Cc )

(25)

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