GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves

International Journal of Food Microbiology 102 (2005) 95 – 105 www.elsevier.com/locate/ijfoodmicro

Short communication

GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves A.H. Geeraerd, V.P. Valdramidis, J.F. Van ImpeT BioTeC-Bioprocess Technology and Control, Department of Chemical Engineering, Katholieke Universiteit Leuven, W. de Croylaan 46, B-3001 Leuven, Belgium Received in revised form 29 September 2004; accepted 1 November 2004

Abstract This contribution focuses on the presentation of GInaFiT (Geeraerd and Van Impe Inactivation Model Fitting Tool), a freeware Add-inn for MicrosoftR Excel aiming at bridging the gap between people developing predictive modelling approaches and end-users in the food industry not familiar with or not disposing over advanced non-linear regression analysis tools. More precisely, the tool is useful for testing nine different types of microbial survival models on user-specific experimental data relating the evolution of the microbial population with time. As such, the authors believe to cover all known survivor curve shapes for vegetative bacterial cells. The nine model types are: (i) classical log-linear curves, (ii) curves displaying a so-called shoulder before a log-linear decrease is apparent, (iii) curves displaying a so-called tail after a log-linear decrease, (iv) survival curves displaying both shoulder and tailing behaviour, (v) concave curves, (vi) convex curves, (vii) convex/concave curves followed by tailing, (viii) biphasic inactivation kinetics, and (ix) biphasic inactivation kinetics preceded by a shoulder. Next to the obtained parameter values, the following statistical measures are automatically reported: standard errors of the parameter values, the Sum of Squared Errors, the Mean Sum of Squared Errors and its Root, the R 2 and the adjusted R 2. The tool can help the end-user to communicate the performance of food preservation processes in terms of the number of log cycles of reduction rather than the classical D-value and is downloadable via the KULeuven/BioTeC-homepage http://cit.kuleuven.be/biotec/ at the topic bDownloadsQ (Version 1.4, Release date April 2005). D 2005 Elsevier B.V. All rights reserved. Keywords: Predictive microbiology; Microbial inactivation; Microbial survival; Freeware tool; Excel add-in

1. Introduction Inactivation of vegetative bacterial cells,1 whether due to a thermal or non-thermal food processing T Corresponding author. Fax: +32 16 32 29 91. E-mail address: [email protected] (J.F. Van Impe). 0168-1605/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2004.11.038

1 Excluding the activation shoulder which can be observed during the inactivation of bacterial spores (see, for example, Sapru et al., 1992).

A.H. Geeraerd et al. / International Journal of Food Microbiology 102 (2005) 95–105

Log10(N)

Log10(N)

96

time

time

Fig. 1. Commonly observed types of inactivation curves. Left plot: linear (5, shape I), linear with tailing (, shape II), sigmoidal-like (5, shape III), linear with a preceding shoulder (o, shape IV). Right plot: biphasic (5, shape V), concave (, shape VI), biphasic with a shoulder (5, shape VII), and convex (o, shape VIII).

technique like high hydrostatic pressure, pulsed electric field, ohmic heating, . . . which are currently under exploration (Anonymous, 2000), can exhibit one of the eight shapes illustrated in Fig. 1 (see, for example, Xiong et al., 1999, Mafart et al., 2002 or Devlieghere et al., 2004). It is clear that the classical concept of loglinear inactivation modelling fails to assess accurately the majority of these survival curves. However, an accurate assessment is important with respect to preventing the design of fail-dangerous, or, on the contrary, overly conservative processing treatments. In this research, the focus is on the development of a user-friendly interface enabling easy identification of (one of) these curvatures on an experimental data set provided by the end-user. After testing several of the possible models included in GInaFiT, the results can be compared based on some reported statistical measures.

available in literature is motivated as follows, combining (general) model criteria as formulated by, amongst others, McMeekin et al. (1993), Van Impe et al. (1992, 1995), Baranyi and Roberts (1994), Rosso et al. (1995) and Geeraerd et al. (2000). Wherever possible, models existing in a dynamic formulation were selected, i.e., written as a (set of) differential equations without the occurrence of the initial population value in the right-hand side. Only such models can deal with realistic time-varying environmental conditions (e.g., temperature) by using the initial population value at the level of starting the numerical integration of the equations (when providing predictions in such timevarying conditions). Moreover, preference is given to

2. Model structures included in GInaFit In total, nine different model types are implemented in the current GInaFiT version (Version 1.4), covering all eight shapes of survival curves illustrated in Fig. 1. The nine different models are grouped in four Menu Items (see Fig. 2), as will be explained in the next paragraphs. For each Menu Item, the shapes of survivor curves covered are indicated. The selection of the models included in GInaFiT within the vast range of modelling approaches

Fig. 2. Screendump of the additional Menu Item GInaFiT and the nine model types available.

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autonomous model types, i.e., without the explicit occurrence of time in the right-hand side of the differential equations. Lastly, the significance of the model parameters and the simplicity of the selected model equations are also considered important, as this simplifies the initial parameter estimation necessary to perform the non-linear parameter optimization. Notwithstanding these mathematical prerequisites, it is the static version of the models which is actually implemented in GInaFiT, because the aim is providing a tool for identifying parameter values on userspecified static experimental data. The dynamic nature of the selected models enables dynamic predictions in possible future versions of the tool or by making use of other software. All static versions are formulated as survivor curves relating log 10(N) with time t, based on the well known logarithmic transformation of microbial data that stabilises the variance (see, for example, Jarvis, 1989) and, as such, enables the use of the classical Sum of Squared Errors criterion (SSE) during parameter optimization. For the concave/convex curves (Menu Item 3, see further), an autonomous dynamic model is, to the authors’ knowledge, not existing until now, and a nonautonomous model with interpretable parameters has been selected. For the most complex shape (i.e., biphasic with a preceding shoulder), covered in Menu Item 4, a new model type is proposed in this research and compared with existing models based on the aforementioned model criteria.

microbial cell density [cfu/mL], k max [1/time unit] the first order inactivation constant and D [time unit] the decimal reduction time. The degrees of freedom used during parameter estimation by GInaFiT are k max and log 10(N(0)).2 In this model it is assumed that all cells in a population have equal heat sensitivity and that the death of an individual is dependent upon the random chance that a key molecule or btargetQ within it receives sufficient heat (Cole et al., 1993). Despite the worldwide use of this model, especially in the canning industry for the so-called b12D processQ of the proteolytic strains (Group I) of Clostridium botulinum spores (ICMSF, 1996), a lot of deviations have been observed (particularly at lower temperatures and for vegetative cells) indicating that inactivation kinetics are not always following first order log-linear relationships (Anonymous, 2000), as was also illustrated in Fig. 1. This observation was at the basis of the development of a number of non-loglinear modelling equations as covered by the remaining GInaFiT Menu Items.

2.1. Menu Item 1: log-linear model covering shape I survivor curves

dCc ¼  kmax d C c : dt

The first Menu Item provides the end-user with the possibility to use the traditional log-linear approach for describing microbial inactivation curves (see, for example, Anonymous, 2000). In static conditions, this can be written as t log10ð N Þ ¼ log10ð N ð0ÞÞ  D kmax t ð1Þ ¼ log10ð N ð0ÞÞ  lnð10Þ

Herein, C c is related to the physiological state of the cells [–], k max is the specific inactivation rate [1/time unit], and N res is the residual population density [cfu/ mL]. The model uses four degrees of freedom: two

Herein, N represents the microbial cell density, expressed in, for example, [cfu/mL], N(0) the initial

2.2. Menu Item 2: log-linear model with shoulder and/ or tailing covering shapes I, II, III and IV survivor curves The model was originally developed as two coupled differential equations, reading as follows (Geeraerd et al., 2000).

 dN 1 Nres ¼  kmax dN d d 1 dt 1 þ Cc N ð2Þ

2 A note about using log 10(N(0)) as a degree of freedom is the following. During the model parameter identification phase (which is the purpose when using GInaFiT), the measured value of the initial population in the user-specified static experimental data set is as error-prone as the other datum points. Therefore, it is not imposed that the model should pass exactly through this point, in other words, it is necessary that log 10(N(0)) uses a degree of freedom in this and all other model types.

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initial states N(0) and C c(0), and two parameter values k max and N res. The first factor at the right-hand side of the dN/dt equation models the log-linear part of the inactivation curve and is equivalent to the classical first-order inactivation kinetics (as in Eq. (1) when written in dynamic format). The second factor describes the shoulder effect and is based on the hypothesis of the presence of a pool of protective or critical components C c around or in each cell (Mossel et al., 1995). Gradually, this pool is destroyed. In case of a shoulder, 1/(1+C c(0)) takes on a small (positive) value. Towards the end of the shoulder region 1/(1+C c(t)) becomes (approximately) equal to one, due to the component C c undergoing first-order heat inactivation. Finally, the last factor of the dN/dt equation implies the existence of a more resistant subpopulation N res, which can be framed within the established mechanistic or vitalistic concepts (Cerf, 1977). For more details about the construction of this model as based on literature arguments (Mossel et al., 1995; Cerf, 1977), reference is made to Geeraerd et al. (2000). It should be noted that the model remains an empirical model, as it is constructed in order to obey a set a predefined requirements and in order to be consistent with literature arguments, but it was not mechanistically derived out of a fundamental study of inactivation phenomena occurring in and around microbial cells. This model can exhibit a log-linear behaviour with and without shoulder and/or tailing revealing a smooth transition between each phase. It is important to remark that, for this model, tailing is considered for a population remaining constant in time or, otherwise stated, not undergoing any significant subsequent inactivation. This is in contrast with the biphasic model types covered in Menu Item 4. The explicit solution of the original dynamic model reads as follows, after substituting C c(0) by ek maxS l1 with S l [time units] a parameter representing the shoulder, as derived in Geeraerd et al. (2000). N ðt Þ ¼ ð N ð0Þ  Nres Þd ekmax t
 ekmax Sl d þ Nres 1 þ ðekmax Sl  1Þdekmax t

ð3Þ

In this formulation, all parameters have a clear biological/graphical meaning and the three phases (shoulder, log-linear phase and tailing region) are easily recognisable.

It should be observed that, when actually applying this model to experimental data, the following format is used h  log10ð N Þ ¼ log10 10log10ðN ð0ÞÞ  10log10ðNres Þ
 ekmax Sl dekmax t d 1 þ ðekmax Sl  1Þdekmax t i þ 10log10ðNres Þ ð4Þ with log 10(N(0)), log 10(N res), k max and S l being the degrees of freedom used for the parameter estimation by GInaFiT. The first two (reduced) models in Menu Item 2 can be derived by setting S l or N res equal to zero in Eq. (3) or (4). The model structure has been successfully applied to survival data of different microorganisms and different treatments, such as Listeria monocytogenes and Lactobacillus sakei during a mild thermal inactivation (Geeraerd et al., 2000), Monilinia fructigena and Botrytis cinerea during a pulsed white light treatment (Marquenie et al., 2003), the Acid Tolerance Response (ATR) of Salmonella enterica and L. monocytogenes (Greenacre et al., 2003), the inactivation of L. monocytogenes in a pH-modified chicken salad during cold storage (Guentert et al., 2003) and the mild temperature inactivation of Escherichia coli K12 (Valdramidis et al., 2003, 2005). 2.3. Menu Item 3: Weibull type models covering shapes I, VI and VIII survivor curves The Weibull model, when applied to describe microbial inactivation (see, for example, Peleg and Cole, 1998; Van Boekel, 2002), is the cumulative form of the asymmetric Weibull probability density function for the heat resistances of individual microbial cells. Other distributions, like the symmetric Fermi distribution, are also possible (Peleg, 2000). It should be noted that this type of modelling is based on a different fundamental view with regard to microbial inactivation in comparison with the models covered in Menu Items 2 and 4: the shoulder is not included a priori as a physical reality, but emerges in an elegant way due to the fact that the distribution reflecting the spectrum of resistances has a large mean relative to its variance (Peleg, 2000).

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In GInaFiT, it is the version as proposed by Mafart et al. (2002) which is included based on the possibility to reduce in a natural way to the classical log-linear model (depending on data behaviour).  t p log10ð N Þ ¼ log10ð N ð0ÞÞ  d Herein, d [time unit] is a scale parameter and can be denoted as the time for the first decimal reduction if p=1, and p [–] is a shape parameter. For pN1, convex curves are obtained, while for pb1, concave curves are described. Van Boekel (2002) and Mafart et al. (2002) observed a strong correlation between the parameters d and p. The dependency of the parameters is due to the model structure (i.e., an error in d will be balanced by an error in p). This drawback can be circumvented by fixing the value of p (see also, e.g., Peleg and Penchina, 2000). Fixing the p-parameter is one of the options under Menu Item 3 of GInaFiT. In this case, a pop-up window appears asking the user which value of p he wants to select. In practice, this option is suitable only when describing a range of different inactivation curves individually (for example, collected at increasing constant temperature values) by using the full Weibull model. Afterwards, if an average p-value seems appropriate, this mean value can be used for all curves (as illustrated in Mafart et al., 2002) by using this additional option under Menu Item 3. The last option under this Menu Item is the model recently proposed by Albert and Mafart (2003, 2005), and which is able to describe concave, convex or linear curves followed by a tailing effect. The model can be written as follows. h i t p log10ð N Þ ¼ log10 ð N ð0Þ  Nres Þd10ðð d Þ Þ þ Nres For identification purposes, the model is written as h  log10ð N Þ ¼ log10 10log10ðN ð0ÞÞ  10log10ðNres Þ i t p  10ðð d Þ Þ þ 10log10ðNres Þ The four degrees of freedom used are d [time unit], p [–], log 10(N(0)) and log 10(N res). Convex/concave curves are widespread, for example, Van Boekel (2002) lists a large number of bacteria and yeasts literature data undergoing a mild heat treatment and showing convex or concave behaviour.

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2.4. Menu Item 4: biphasic models covering shapes I, II, V and VII survivor curves Cerf (1977) proposed a two-fraction model, which can be formulated as follows log10ð N Þ ¼ log10ð N ð0ÞÞ   þ log10 f d ekmax1 t þ ð1  f Þd ekmax2 t ð5Þ Herein, f is the fraction of the initial population in a major subpopulation, (1f) is the fraction of the initial population in a minor subpopulation (which is more heat resistant than the previous one), and k max1 and k max2 [1/time unit] are the specific inactivation rates of the two populations, respectively. This biphasic model is capable of describing survivor curves of shapes I, II and V, and can, if being used for time-varying conditions, be written under the form of two first-order differential equations, one for N 1 (the major subpopulation) and one for N 2 (the minor subpopulation). Biphasic inactivation curves have been observed in the framework of thermal inactivation (see, e.g., Cerf, 1977; Humpheson et al., 1998) and non-thermal inactivation due to lethal water activity (Shadbolt et al., 1999) or lethal pH levels (Shadbolt et al., 2001). Zhang et al. (1994), amongst others, support biphasic pulsed electric field inactivation curves. Survivor curves of shape VII (i.e., biphasic with a preceding shoulder) are rather uncommon. Whiting (1993) modelled the survival of L. monocytogenes and three Salmonella strains in BHI broth under stressful conditions of combined lactic acid, NaCl and/or NaNO2 at temperatures between 4 and 42 8C. Most of the curves displayed shape IV, but some of the curves were shape VII. Similar observations were made by Whiting et al. (1996) for the non-thermal inactivation of Staphylococcus aureus. In order to describe survivor curves of shape VII a new model structure is proposed, combining interesting features of the Geeraerd et al. (2000) model, as presented under Menu Item 2, and the Cerf model (Eq. (5)). In dynamic conditions, the model reads as follows.
 dN1 1 ¼  kmax1 dN1 d 1 þ Cc dt

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 dN2 1 ¼  kmax2 d N2 d 1 þ Cc dt dCc ¼  kmax1 d C c dt As in the Cerf model, the total population N [cfu/ mL] equals the sum of the two subpopulations N 1 and N 2, and a fraction f is defined as N 1(0)/N(0). The major population is more sensitive (k max1 is larger than k max2) and a shoulder is introduced similarly as in Eq. (2). For static conditions, the model can be written as log10ð N Þ ¼ log10ð N ð0ÞÞ þ log10
   f d ekmax1 t þ ð1  f Þd ekmax2 t  ekmax1 Sl d 1 þ ðekmax1 Sl  1Þdekmax1 t Herein, the degrees of freedom used are log 10(N(0)), f, k max1, k max2, and S l. Again, all parameter values have a clear significance and the different segments of the survivor curve (shoulder, biphasic pattern) are easily recognisable. If S l=0 (after identification on experimental data), the model reduces in a natural way to the Cerf model. In literature, two modelling approaches are already available capable to describe shape VII survivor curves, but were not selected for inclusion in GInaFiT. This is motivated as follows. 1. Richard Whiting (1993) proposed a logistic-based model including a shoulder and two populations. In his formulation, a parameter tl appears, related with the shoulder period. However, a model without a shoulder can only be correctly attained by selecting tl=l, which is not a realistic biological value. On the contrary, in the new model proposed in the present research, a shoulder is not present when S l=0. Also, when deriving the dynamic version of the Whiting model, it is not possible to eliminate both N(0) and t (as analysed in Geeraerd et al., 2000). 2. Xiong et al. (1999) proposed an extension of the Buchanan et al. (1993) inactivation model. The model is inherently static (i.e., formulated with different equations for different time segments

before and after the completion of the shoulder period) and cannot be used for time-varying environmental conditions.

3. Statistical measures included in GInaFiT The following statistical measures are automatically reported for each model selected by the user. 1. The Sum of Squared Errors (SSE), obtained by summing the squared differences between the experimental data and the identified model, both in log 10-scale. 2. The Mean Sum of Squared Errors (MSE), which can be derived by dividing SSE by the number of degrees of freedom nk, i.e., the number of data points n minus the number of degrees of freedom k (parameters and initial values) used (two, three, four or five; depending on the model structure). 3. The Root Mean Sum of Squared Errors (RMSE), the square root of MSE. As elaborated upon by Ratkowsky (2003), bthe RMSE can be considered as the most simple and most informative measure of goodness-of-fit, both for linear and non-linear modelsQ. The magnitude of RMSE should be comparable to the standard deviation (as a measure of the precision of replicates) of the experimental data. When the magnitude of RMSE is much larger than the experimental precision, a more accurate model is needed, for example, using more degrees of freedom. On the contrary, when the magnitude is much smaller, the model is overfitting the data, i.e., following the noise instead of the general trend in the data and a less complex model is needed, for example, using less degrees of freedom. 4. The two, three or four parameter values identified, together with their Asymptotic Standard Error. This Standard Error is obtained by taking the square root of the diagonal elements of the asymptotic variance–covariance matrix AVC, with AVC= MSEd ( J T d J)1. Herein, J corresponds to the Jacobian matrix containing the partial derivatives of the model output with respect to the model parameters evaluated at each experimental time instant. Generally, an underestimation of the actual confidence range of the parameter estimates by using

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Standard Errors for a non-linear model is to be expected as covariance terms are neglected. More appropriate measures for non-linear models include the construction of joint confidence regions (Beale, 1960) or a Monte Carlo analysis based on knowledge of the experimental error (see, for example, Poschet et al., 2003). These measures are not included in the present version of GInaFiT. 5. R 2, the coefficient of determination which equals 1SSE/SSTO, with SSTO the sum of the squared differences between the measured values and the mean of these measured values. 6. R 2adj, the adjusted coefficient of determination (see, for example, Wonacott and Wonacott, 1990), 2 kþ1 which equals R2adj ¼ ðn1ÞdR . Again, n equals nk the number of data points and k the number of degrees of freedom used. In comparison with the original R 2, the R 2adj attempts to penalize the inclusion of an irrelevant variable, in case, a redundant parameter. However, as exemplified by Ratkowsky (2003), RMSE is more suitable for non-linear models.

4. Programming issues For the development of this tool MicrosoftR Excel was chosen based on three criteria: (i) Microsoft Office is commonly available on most personal computers, (ii) MS-Visual Basic offers a large flexibility for automating the process and developing the user interface, and (iii) the Solver Add-In for the non-linear parameter estimation of the selected model structures is available. In order to make the installation as userfriendly as possible the tool is written as an Add-In for MS-Excel available to the user at any time as an additional Menu Item (as already shown in Fig. 2). In order to enhance convergence of the non-linear parameter optimization, GInaFiT has some built-in logical constraints on the parameter values to be estimated, for example, k max N0, S l N0 and log 10(N res)blog 10(N(0)) for the Geeraerd et al. model; 0bfb1 and k max1Nk max2N0 for the Cerf model and the new model proposed in the present research. GInaFiT will deliver error messages when parameters which are fairly unlikely or unreliable are obtained, for example, a negative shoulder length or an estimated value of the residual population N res

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lower than the smallest measured population value. For the latter case, this error message states that the user should prefer a model without tailing, as tailing seems not to be substantiated by his data.

5. An illustrative case study on the thermal inactivation of E. coli K12 5.1. Data generation Survival data of early stationary phase cultures of E. coli K12 MG1655, a surrogate for the food-borne pathogen E. coli O157:H7 are collected (Valdramidis et al., 2003, 2005). The selected case study takes place in BHI broth in capillaries immersed in a constant temperature circulating water bath at 56.6 8C (GR150S12, Grant). Cell density is determined by plate counting on BHI. 5.2. Importing data into GInaFiT and model application In an open MS-Excel sheet the experimental data consisting of t (first column) and log 10(N) (second column) are selected. Next, based on the model(s) pointed at in the GInaFiT Menu Items, the tool performs the parameter estimation minimising the Sum of Squared Errors criterion. 5.3. GInaFiT model outputs The selected case study data are fitted with eight out of the nine models of the tool, excluding the Weibull model with fixed p as this model is suitable only when describing a range of experimental data sets. As an example, the GInaFiT output for model 2 (Log-Linear+Shoulder) is shown in Fig. 3. As can be seen, the model equation and its version used during parameter optimization are reported in the result sheet. To summarize the results of the eight models, the SSE, RMSE and obtained parameter values are reported in Table 1 (for the sake of comparison of the obtained parameter values and in order to comment on the GInaFiT warnings, even models which are obviously not suitable for these clearly curved data are included in this table). The following important observations are to be made.

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Fig. 3. GInaFiT output when selecting the bGeeraerd et al. (2000): Log-Linear+ShoulderQ Menu Item for the case study at hand.

Firstly, for the models with tailing (3, 4, and 7), the program warns that this tailing is not substantiated by the data and that a model without tailing should be

selected. Convergence for the residual population value N res is still obtained (at a very low level with an unrealistic high Standard Error (not shown), which

Table 1 Statistical measures and parameter values obtained when applying eight models available in GInaFiT Version 1.4 on the experimental data set at hand Model type

SSE

RMSE

Log 10(N(0)) [cfu/mL]

k max [1/min]

Menu Item 1 (1) Log-linear regression

3.95

0.57

10.33

0.51

Menu Item 2 (2) Log-linear+shoulder (3) Log-linear+tail (4) Log-linear+shoulder+tail

0.33 3.95 0.33

0.17 0.60 0.18

9.47 10.33 9.47

0.71 0.51 0.71

Menu Item 3 (5) Weibull (7) Weibull+tail

0.55 0.55

0.22 0.24

9.63 9.63

Menu Item 4 (8) Biphasic model (9) Biphasic+shoulder

3.95 0.33

0.63 0.19

10.33 9.47

n.s.=non-significant (see text).

Sl [min]

Log 10(N res) [cfu/mL]

d [min]

p [–]

10.03 10.03

1.80 1.80

f [–]

k max1 [1/min]

k max2 [1/min]

0.00n.s 0.77n.s.

2.92n.s. 0.71

0.51 0.71

8.63 8.63

3.04n.s. 1.10n.s.

4.08n.s.

8.63

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also indicates the redundancy), while the other parameter values are identical to the ones obtained by models 1, 2 or 5, respectively. Secondly, the biphasic model fit (model 8) is actually equal to the log-linear model fit (model 1) as reflected in identical values for log 10(N(0)) and k max2. The unnecessary f parameter is estimated as being zero, and a numerical error is indicated for calculating the Standard Errors for the biphasic model. Also, the SSEs of models 1 and 8 are identical, while the inclusion of the irrelevant parameter values is punished by a higher RMSE for model 8. Selfevidently, model 9 is also not suitable for this case study, as it is even more complex than model 8. Nevertheless, convergence of the parameter estimation procedure is still attained (with an unrealistic high Standard Error for f (not shown), which also indicates the redundancy). Observe that the other parameter values are identical to those of model 2. Thirdly, the model description is more accurate for the (remaining) models 2 and 5 in comparison with model 1, as reflected in the SSE and RMSE values. In this case, the RMSE of the simpler model is higher, as model 1 has a very large SSE. Finally, models 2 and 5 lead to similar RMSE values of 0.17 and 0.22, respectively, and the models seem to describe accurately the data. As no true (independent) replicates of the case study are available, a careful comparison of the RMSEs and the experimental variation as a basis for model discrimination is not feasible (an example is reported in Ratkowsky, 2003). However, an indication can be found in Mossel et al. (1995), reporting a standard deviation of 0.25–0.5 for regularly standardized bacterial colony counts, whereas for rigorously standardized bacterial colony count a standard deviation of 0.10–0.15 can be assumed. It should be noted that these values deal only with the measurement variation, which is a (probably minor) subpart of the total experimental variation. Accepting this limitation, the obtained RMSEs both for models 2 and 5 appear acceptable.

6. Concluding remarks GInaFiT Version 1.4 is useful to quickly test in a quantitative and user-friendly way a range of microbial survival models on user-specific data. To the best

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of the authors’ knowledge, all survivor curve shapes observed until now in literature for vegetative cells are covered. Next to the derivation of parameter estimates and standard errors, some goodness-of-fit indicators are presented as well. The tool can help the end-user to communicate the performance of food preservation processes making use of the concept of the number of log cycles of reduction, rather than the classical Dvalue. This concept originates from Buchanan et al. (1993) and was recently incorporated in the Recommendations for Further Research at the second Research Summit of the Institute of Food Technologists in Orlando, January 14–16, 2003 (Heldman and Newsom, 2003). The software, which is currently available for Office97 (English and French), Office2000 (English) and OfficeXP/2002 (English), is distributed under a freeware license agreement. If being asked for, the GInaFiT code can easily be adapted in order to be suitable for other Office languages. In the future, an extension towards the influence of several (possibly interacting) environmental and processing factors like temperature, pH, water activity, recovery conditions, food structure and composition, . . . on the shape and extent of the microbial survival curves is necessary. At the moment, some secondary models for relating the effect of temperature, pH and/ or water activity on microbial inactivation parameters are available (for example, Cerf et al., 1996), but more research is needed, especially on the effect of food structure and composition. This step would enable food producers to have an overall view on the influence of their processing conditions on microbial survival, and hence, to assess in an accurate way their processes performances in the framework of, for example, tackling Food Safety Objectives.

Acknowledgements Arnout Standaert, Mieke Janssen, and Karen Vereecken (KULeuven/BioTeC) are gratefully acknowledged for their help in testing (earlier) versions of this tool, as well as Marie Cornu (AFSSA, France) for her help to establish the French version. Research at BioTeC in predictive modelling in foods is supported by the Research Council of the Katholieke Universiteit Leuven as part of projects IDO/00/

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