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Journal of Applied Microbiology 1999, 87, 491–499

Validating predictive models of food spoilage organisms C. Pin, J.P. Sutherland1 and J. Baranyi Institute of Food Research Reading Laboratory, Earley Gate, Reading, UK 7114/03/99: received 5 March 1999, revised 14 May 1999 and accepted 18 May 1999

The accuracy and bias of a predictive model for the maximum specific growth rate of Pseudomonas spp. were studied by means of percentage discrepancy and bias indicators. These were calculated for observations obtained both in laboratory media and in food. When independent pseudomonad data generated in broth were compared with model predictions, the error was smaller than in the case of food. The extent to which the food structure and composition of the microflora contribute to the overall error of the model was quantified. C . P IN , J. P. S UT HE R LA ND A ND J. B AR AN Y I. 1999.

INTRODUCTION

Numerous publications have described the construction of predictive models for the growth of food-borne pathogens (Guerzoni et al. 1994; Bhaduri et al. 1995; Sutherland et al. 1997; McClure et al. 1997) and food spoilage organisms (Abdullah et al. 1994; Kalathenos et al. 1995; Neumeyer et al. 1997a; Pin and Baranyi 1998). Some authors have also attempted to validate models in foods and to discuss the sources and types of error associated with validation (Ross 1996; Neumeyer et al. 1997b; Baranyi et al. 1999). It is important to improve the accuracy and representational value of the models as much as possible, in order that they give realistic predictions without being excessively ‘fail-safe’. This will promote confidence in the value of predictive models within the food industry and encourage their application in real situations, with consequent economic benefit. Predictions from spoilage models based on observation of microbiological responses in a well-defined and controlled laboratory environment, using microbiological media, are not necessarily valid in a real food environment, which is highly complex and sometimes not easily quantifiable (e.g. the structure of food). Frequently, it is difficult to decide on the degree to which the complexity of the food environment can be discounted. Suppose that, for a given organism, a predictive model was created by fitting the model to data obtained in broth, under laboratory conditions. The bias and inaccuracy of the model when predicting bacterial responses under similar laboratory conditions will subsequently be called primary bias and priCorrespondence to: C. Pin, Departamento de Nutricio´n y Bromatologı´a III, Facultad de Veterinaria, Universidad Complutense de Madrid, 28040 Madrid, Spain. 1 Present address: School of Health and Sports Science, Faculty of Science, Computing and Engineering, University of North London, 166–220 Holloway Road, London N7 8DB, UK. © 1999 The Society for Applied Microbiology

mary error. When the model is applied to natural food conditions, the bias and inaccuracy of the model will be called overall bias and overall error. The two kinds of errors are schematically represented in Fig. 1. Baranyi et al. (1999) introduced the percentage bias and discrepancy for a model which we will use, in this paper, to quantify the primary and overall bias and error. While, for a reasonable model, the percentage primary bias is expected to be close to zero, the percentage overall bias may be significantly different from zero, because of the differences between the broth and food environments. We call the bias positive or negative according to whether the model over- or underestimates the bacterial growth. Since the overall error includes the variability inducing the primary error, any quantification of the overall error should be greater than that of the primary error. The differences between the overall and primary error characterize the extent to which the laboratory environment represents the food environment. If over- (or under-) estimations were the main reason for the inaccuracy of the model, then the percentage overall bias is close to the percentage overall error (or discrepancy). If the model is ‘good enough’ (this depends on the quality of the raw data, robustness and qualitative features of the model; see Baranyi and Roberts 1995), then the primary error can be satisfactorily estimated by mathematical and statistical means, assuming that the data on which the model is based (ž in Fig. 1) are sufficiently representative. The overall bias and error can be measured by comparing the predicted values for a specific parameter given by the model with those observed in food. In this work, the model of Pin and Baranyi (1998) for the maximum specific growth rate of Pseudomonas spp. is extended. The primary bias and error of the model are studied

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Fig. 1 As the system, to which a

predictive model is applied, becomes more complex, the error of the predictions increases

by comparing growth rates predicted by the model and observed under laboratory conditions in microbiological media. The overall bias and error are measured by comparing growth rates predicted by the model and observed in food allowed to spoil ‘naturally’ (milk and meat). The main differences between laboratory and ‘real’ (food) conditions are the growth substrate and microbial diversity of the food. To eliminate the contribution of the latter, pseudomonads were inoculated into ultra heat-treated (UHT; commercially sterile) milk and irradiated meat. In this case, the inaccuracy of the model will be called intermediate error. This error is expected to be greater than the primary error, because of the contribution of the variability of the substrate (food vs broth), but smaller than the overall error because of the control over the bacterial flora (controlled vs wild). The increase in error, from primary to overall, reflects the increase in complexity of the system, from sterile broth inoculated with a single organism to non-sterile food with a diverse microflora (Fig. 1). The results of the modelling experiments reported here will add to our knowledge and understanding of the way in which the growth of micro-organisms is affected by each other and by the food in which they are growing. MATERIALS AND METHODS Media

Tryptone Soya broth (TSB; CM129; Oxoid, Unipath Ltd, Basingstoke, UK) was adjusted to the target pH values using 5 mol l−1 HCl, dispensed in 250 ml volumes and autoclaved at 121 °C for 15 min. Bacteria were enumerated on Tryptone Soya agar (TSA; CM 131; Oxoid) for non-selective (total viable) counts and

Cetrimide Fusidin Cephaloridine agar (CFC; CM559 with added supplement SR 103; Oxoid) for Pseudomonas spp. Bacterial strains

Thirty-eight strains of Pseudomonas, Acinetobacter, Psychrobacter, Shewanella, Enterobacteriaceae, Carnobacterium, Lactobacillus, Leuconostoc, Bacillus spp., Brochothrix thermosphacta and Kurthia spp., were grouped as follows. Group P. Pseudomonas putida National Collection of Food Bacteria (NCFB) 754 (spoiled milk), Ps. fragi NCFB 2902 (beef), Ps. lundensis NCFB 2908 (minced beef), Shewanella putrefaciens NCFB 756 (tinned butter), Acinetobacter sp. National Collection of Industrial and Marine Bacteria (NCIMB) 11168 (lamb carcass meat), Acinetobacter sp. NCIMB 11169 (lamb carcass meat), Psychrobacter inmobiliz NCIMB 11372 (pork sausage) and P. immobiliz NCIMB 11650 (meat). Group T. Including group P and Enterobacter agglomerans NCFB 2071 (minced meat), Ent. agglomerans NCFB 2073 (pasteurised milk), Klebsiella oxytoca NCFB 2678 (Cheddar cheese), Kl. pneumoniae NCFB 1010 (unknown), Escherichia coli NCFB 555 (raw milk), Proteus morganii NCTC 235 (unknown), Lactobacillus sp. NCFB 2812 (vacuum-packed pork), Lactobacillus sp. NCFB 2813 (vacuum-packed beef), Lactobacillus sp. NCFB 2814 (vacuum-packed bacon), Lactobacillus sp. NCFB 2815 (vacuum-packed lamb), Leuconostoc carnosum NCFB 2775 (vacuum-packed meat), Leuc. gelidum NCFB 2776 (vacuum-packed meat), Leuc. gelidum NCFB 2800 (vacuum-packed beef), Carnobacterium divergens NCFB 2856 (vacuum-packed beef), C. divergens NCFB 2857 (vacuum-packed lamb), C. piscicola NCFB 2853 (vacuum-packed beef), C. piscicola NCFB 2854 (vacuum-packed lamb), Broc. thermosphacta NCFB 1676 (fresh pork sausage), Broc. ther-

© 1999 The Society for Applied Microbiology, Journal of Applied Microbiology 87, 491–499

P RE DI C TI VE M OD EL S VA LI D AT IO N 493

mosphacta NCFB 2830 (lamb), Broc. thermosphacta NCFB 2849 (vacuum-packed pork), Kurthia gibsonii NCIMB 10499 (pork sausage), K. zopfii NCIMB 10494 (hamburgers), K. zopfii NCIMB 10496 (frozen mince pork), K. zopfii NCIMB 10498 (pork sausage), K. gibsonii NCIMB 10495 (fat trimmings), K. gibsonii NCIMB 10497 (rendered lard). Group Mk. Of similar composition to group T, except for Broc. thermosphacta and Kurthia spp. strains which were substituted by Bacillus cereus NCFB 577 (milk), B. cereus NCFB 578 (milk), B. coagulans NCFB 1761 (evaporated milk), B. coagulans NCFB 1211 (sterilized milk), B. coagulans NCFB 1210 (sterilized milk) and B. subtilis NCFB 1069 (bulk tanker milk). Inoculum preparation

Active growth of each strain was ensured by three subsequent 24-h subcultures in 10 ml TSB incubated at 25 °C during the 3 d preceding the start of the experiments. Equal volumes of the final subcultures, diluted to give approximately 105 cfu ml−1, were combined to give the cocktails of the groups described above.

lated with group T were plated on CFC and TSA. Plates were incubated at 25 °C for 24 h.

Sterile food experiments

Minced beef obtained from a local wholesaler was sterilized by irradiation (10 Kgray). Ultra heat-treated milk was purchased from a local shop. The sterile minced beef was inoculated with groups P and T (approximately 103 cfu g−1). Milk was inoculated with groups P and Mk (approximately 103 cfu ml−1). After inoculation and thorough mixing, the minced beef was divided into units of 5 g, which were packaged in sterile polythene bags and sealed. The inoculated milk was aseptically dispensed in 10-ml volumes into sterile bottles. The pH, aw and initial bacteria number were determined from one of the samples. The meat and milk samples were incubated at 2, 5, 8 and 11 °C and milk was additionally incubated at 16 °C, with sampling at recorded times to generate data for growth curves. For samples inoculated with group P, bacteria were recovered on TSA. Bacteria from foods inoculated with groups T and Mk were recovered on TSA and CFC. Plates were incubated at 25 °C for 24 h.

Broth experiments

Tryptone soya broth (250 ml) was inoculated with group P and group T to give final concentrations of approximately 103 cfu ml−1. After inoculation, each broth was aseptically dispensed in 10-ml volumes into sterile bottles and incubated at the desired temperature for data generation (2, 5, 8, 11 and 16 °C) at pH 6·8 (see Fig. 2). During incubation, samples inoculated with group P were plated on TSA. Samples inocu-

‘Natural food’ experiments

Data obtained from naturally spoiling (uninoculated, untreated) minced beef (Baranyi et al. 1999) were used in this work. Pasteurised milk was purchased from a local shop and aseptically dispensed in 10 ml amounts into sterile bottles. The pH, aw and initial bacterial number were estimated using one of the samples and the remaining samples incubated at 5, 10 and 15 °C. The meat and milk were sampled at suitable intervals and bacteria were recovered on CFC and TSA incubated at 25 °C for 48–72 h.

Growth models

Fig. 2 Interpolation region of the model Me(G). ž, Conditions at which growth was observed. Thick frame, Boundary of the interpolation region of the model M(G) of Pin and Baranyi (1998). Thin frame, Boundary of the interpolation region of the extended model Me(G) containing that of M(G)

The sigmoid growth curves of Baranyi and Roberts (1994) were fitted to the data generated. Pin and Baranyi (1998) described a generic model, denoted M(G), for the maximum specific growth rate of the fastest growing spoilage bacteria using the data of the dominant organisms. The model was a standard multivariate second order polynomial for Ln m, the natural logarithm of the maximum specific growth rate. It was found that M(G) did not describe the observed growth rates in the additional region of temperature and pH accurately (see Fig. 2). Hence, a new quadratic response surface for Ln m was generated, based partly on the data of Pin and Baranyi (1998) and partly on the new data obtained in the additional environmental region. Following the pro-


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cedure of that paper, three models were created in the first step: 1 Me(P,TSA), extended model of group P, grown in isolation, using counts from TSA plates; 2 Me(T,CFC), extended model of group T, using counts from CFC plates; 3 Me(T,TSA), extended model of group T, using counts from TSA plates. Comparison of the three models using the F-test revealed no significant difference between them (P × 0·05); therefore an extended generic model Me(G) was created from the three groups of data. Also, the F-test was used to check whether the contribution of the pH as an explanatory variable in the model MeG) was significant. The environmental region, within which the predictions of the model Me(G) are interpolated values, is referred to as the minimal convex polyhedron (MCP or convex hull) as described by Baranyi et al. (1996).

Estimating the primary and overall errors

Following the definitions of Baranyi et al. (1999), consider x [x1, x2] as a two-dimensional random variable where x1 temperature, x2 pH and suppose that x is uniformly distributed in the environmental region. The accuracy of an f(x) predictive model, describing the m(x) maximum specific growth rate of an organism as a function of the environmental vector x is defined by A exp (zE[(Ln f (x) − Ln m(x))2]) where E(.) denotes the expected value of a random variable. The percentage discrepancy (or error) between the model and observation is %D (A − 1)·100% With the assumption that the f(x) predictive model was created by fitting data obtained in broth inoculated with the organism in question, if the maximum specific growth rate observations were made under similar conditions, then %D characterizes the primary error. It refers to the overall error, if the observations were made in food with its natural microflora. Similarly, the bias factor is defined by B exp (E[Ln f (x) − Ln m(x)]) from which the percentage bias is %B sgn (Ln B) · (exp =Ln B= − 1) · 100% If m observations, m(1), m(2), . . . m(m), are given randomly in the environmental region, then the accuracy and bias factors for f can be estimated by

c

F m (k) (k) 2J G s (Ln f(x ) − Ln m ) G k=1 G Af exp G m j f Fm (k) (k) J G s (Ln f(x ) − Ln m )G k=1 G Bf exp G m f j (for details, see Baranyi et al. 1999). As can be checked, the value of Af cannot be smaller than 1 and it is a certain, non-arithmetical mean of the factors by which predictions and observations differ (considering always the larger of the ratios f(x)/m and m/f(x) for that factor). Therefore, %D expresses an average relative deviation of the measured specific rate from the predicted. When referring to the ‘error of prediction’ by percentage, we mean it in the sense described above. The bias and error of the extended generic model, Me(G), will be quantified by the indicators obtained by the above formulae. For the primary bias and error, the raw data used to create the model Me(G) and that of Neumeyer et al. (1997a), on the growth of pseudomonads in laboratory media, were used. The latter data set consisted of 36 observations from 2·2 to 15·6 °C at 0·996 water activity. The pH for these data was assumed to be 6·8, the upper limit of our pH region. In order to estimate an intermediate error, pseudomonads were inoculated in sterile milk and meat. To calculate the overall bias and error, data obtained from non-sterile milk (published in this paper) and meat (from Baranyi et al. 1999) were used.

RESULTS

The new data obtained from broth are shown in Table 1. No noticeable difference was found between the growth rates of either group P grown in isolation or grown in combination with the bacteria of group T, or the growth of group T plated on TSA, i.e. Me(P,TSA) ¼ (Me(T,CFC) ¼ (Me(T,TSA). Group P grew in the same way in isolation as it did when combined with the other bacteria, when the experiments were carried out in sterile food (Table 2). The results obtained from food, either sterile (inoculated with group T or Mk) or with its natural microflora, showed that the micro-organisms able to grow on CFC were the main group responsible for the total growth recovered on TSA (Tables 2 and 3). The error of the predictions of the model M(G) of Pin and Baranyi (1998), in the environmental region where the model was created (temperature from 2 to 11 °C and pH from 5·2 to 6·4), was about 10–12%. However, when that model was used for conditions out of the convex hull (MCP) of the model (Fig. 2), the error was higher than 60%. This is why



Table 1 Maximum specific growth rates (m) and their S.E. of organisms, belonging to groups P and T, when inoculated in broth

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — Group P on TSA Group T on CFC Group T on TSA (pseudomonads in (pseudomonads selected (total mixture without isolation) from the total mixture) selection) — –––––––––––––––––––––––––– — –––––––––––––––––––––––––– — –––––––––––––––––––––––––– Temp (°C) pH m S.E. m S.E. m S.E. –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — 2 6·8 0·090 0·008 0·071 0·006 0·081 0·009 5 6·8 0·126 0·010 0·121 0·007 0·171 0·018 8 6·8 0·210 0·015 0·165 0·022 0·200 0·027 11 6·8 0·324 0·021 0·287 0·015 0·286 0·012 16 6·8 0·398 0·058 0·349 0·026 0·393 0·050 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — TSA, Tryptone soya agar; CFC, cetrimide fusidin cephaloridine.

Table 2 Maximum specific growth

rates (m) and their S.E. in sterile meat inoculated with groups P and T and in sterile milk inoculated with groups P and Mk

— ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Group P Group T on CFC Group T on TSA — ––––––––––––– –––––––––––––––– — — ––––––––––––––––– Temp (°C) pH m S.E. m S.E. m S.E. — ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Sterile meat 2 5·8 0·066 0·004 0·063 0·003 0·054 0·003 5 5·8 0·109 0·008 0·073 0·004 0·067 0·004 8 5·8 0·206 0·019 0·158 0·012 0·147 0·014 11 5·8 0·301 0·033 0·252 0·011 0·253 0·018 — ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Group P Group Mk on CFC Group Mk on TSA — ––––––––––––– –––––––––––––––– — — ––––––––––––––––– Temp (°C) pH m S.E. m S.E. m S.E. — ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Sterile milk 2 6·7 0·055 0·003 0·098 0·007 0·112 0·014 5 6·7 0·073 0·004 0·124 0·009 0·117 0·010 8 6·7 0·160 0·007 0·190 0·013 0·206 0·035 11 6·7 0·270 0·016 0·277 0·015 0·272 0·013 16 6·7 0·347 0·020 0·330 0·020 0·316 0·047 — ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– CFC, Cetrimide fusidin cephaloridine; TSA, tryptone soya agar.

it was necessary to create the new model, Me(G), by combining the data of Pin and Baranyi (1998) with the new data presented in this paper. Table 4 shows the coefficients of the extended generic model, Me(G), for the Ln m values. The standard error of the fit was 0·15 and the R2 percentage was 93%. The interpolation region of the new, extended model (see Fig. 2) can be described by the inequalities: pH ¾ 6·4;

pH − 5·2; pH − ((6·8–5·2)/(16–11))* (Temp-11) ¦ 5·2; Temp − 2. The significance of the pH as an explanatory variable in the new model was evaluated by an F-test. The probability associated with the F-value was 0·057. It was decided not to exclude pH from the model. The percentage bias of the model with respect to the laboratory observations of Neumeyer et al. (1997a) was close


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Table 3 Maximum specific growth rates (m) and their S.E. in

uninoculated pasteurised milk plated on cetrimide fusidin cephaloridine (CFC) and tryptone soya agar (TSA) — ––––––––––––––––––––––––––––––––––––––––––––––––––––– CFC TSA — –––––––––––––––– –––––––––––––––– — Temp (°C) pH m S.E. m S.E. — ––––––––––––––––––––––––––––––––––––––––––––––––––––– 5 6·7 0·076 0·005 0·058 0·004 10 6·7 0·182 0·013 0·130 0·008 15 6·7 0·299 0·030 0·301 0·018 — –––––––––––––––––––––––––––––––––––––––––––––––––––––

observations in sterile food inoculated with the organisms, we get an intermediate error. Its percentage discrepancy indicator was calculated as 26·6% (Table 6) which is more than double the primary error. This demonstrates that the increased complexity of the growth substrate (food instead of broth) markedly contributes to the overall error of the model (Fig. 3). The only difference between the sterile food experiments and the natural food experiments was the bacterial population (controlled vs wild). Figure 3 demonstrates how this increase in the variability of the microflora changed the error of predictions from 26·6% (intermediate error) to 53·5% (overall error).

Table 4 Coefficients and associated S.E. of the model Me(G) for

the natural logarithm of the maximum specific growth rate of pseudomonads — ––––––––––––––––––––––––––––––––––––––––––––––––––––– Coefficient S.E. — ––––––––––––––––––––––––––––––––––––––––––––––––––––– Intercept −3·929 2·8510 Temperature 0·185 0·0569 pH 0·2142 0·9554 Temperature2 −0·003181 0·0014 pH2 −0·008647 0·0796 Temperature × pH −0·001372 0·0102 — –––––––––––––––––––––––––––––––––––––––––––––––––––––

Table 6 Intermediate bias and error of Me(G) quantified by per

cent bias (%B) and per cent discrepancy (%D) using observations in meat milk with their natural microflora — ––––––––––––––––––––––––––––––––––––––––––––––––––––– %B %D — ––––––––––––––––––––––––––––––––––––––––––––––––––––– Inoculated sterile meat 11·28% 27·57% Inoculated sterile milk 6·96% 25·70% Intermediate bias and error 9·10% 26·64% — –––––––––––––––––––––––––––––––––––––––––––––––––––––

to zero (−0·58%; see Table 5), confirming that the model Me(G) is unbiased (Fig. 4). The percentage discrepancy between Me(G) and the actual data used to create the model was 15·3%. Interestingly, this is higher than the 7·3% discrepancy between Me(G) and the independent data of Neumeyer et al. (1997a), which were obtained under similar laboratory conditions (Table 5). Therefore, the primary error of the model can be expected to be between the two figures; exact calculations showed 11·2%. The only difference between the media experiments and the sterile food experiments was the growth substrate (broth vs food). When comparing the model predictions and the

Table 5 Primary bias and error of Me(G) quantified by per cent bias (%B) and per cent discrepancy (%D) using observations on broth — ––––––––––––––––––––––––––––––––––––––––––––––––––––– %B %D — ––––––––––––––––––––––––––––––––––––––––––––––––––––– Data used to create model Me(G) 0·00% 15·27% Data of Neumeyer et al. (1997a) −0·58% 7·33% Primary bias and error −0·29% 11·23% — –––––––––––––––––––––––––––––––––––––––––––––––––––––

Fig. 3 The primary error of Me(G), i.e. the error between the

model and laboratory observations, is quantified by a per cent discrepancy of 11·2%. These indicators for the intermediate error (predictions of Me(G) compared with observations in inoculated sterile food) and for the overall error (predictions of Me(G) compared with observations in naturally spoiled food) are 26·6 and 53·5%, respectively



Fig. 4 Natural logarithm of the specific growth rate of

pseudomonads, as predicted by the extended model Me(G) vs observed values. ž, Comparisons with the data to which the model was fitted (primary error). Ž, Comparisons with the broth data of Neumeyer et al. (1997a) (primary error). , Comparisons with data, obtained in sterile food, published in this study (intermediate error). , Comparisons with data, obtained in sterile food, published by other authors (intermediate error). ¦, Observations in food, with its natural microflora, published in this work (overall error). ×, Observations in food, with its natural microflora, published by other authors (overall error)

The overall error of Me(G) was 46·3% when applied to meat and 61·17% when applied to milk (Table 7). The difference between the respective figures was appreciably smaller in the case of the intermediate errors: 27·5% for sterile meat and 25·7% for sterile milk (Table 6). Tables 6 and 7 show positive bias, which means that, dominantly, the predictions of the model fail safe. DISCUSSION

The growth rates obtained on both TSA and CFC support the results of Neumeyer et al. (1997b) and Pin and Baranyi

Table 7 Overall bias and error of Me(G) quantified by per cent bias (%B) and per cent discrepancy (%D) using observations in sterile meat and milk — ––––––––––––––––––––––––––––––––––––––––––––––––––––– %B %D — ––––––––––––––––––––––––––––––––––––––––––––––––––––– Meat with its natural microflora* 35·5% 46·3% Milk with its natural microflora 54·2% 61·17% Overall bias and error 44·5% 53·56% — ––––––––––––––––––––––––––––––––––––––––––––––––––––– * Data from Baranyi et al. (1999).

(1998) on the dominance of Pseudomonas spp. in inoculated and uninoculated food as well as in broth when stored under aerobic conditions and refrigeration. Baranyi et al. (1996) emphasized the importance of the correct definition of the interpolation region of a multivariate growth model; in the case of non-mechanistic models, predictions are reliable only if made within the convex hull (MCP) of those points of the experimental design which provided the data to which the model was fitted. Giving the correct interpolation region (and not merely the range of variables) completes the description of an empirical model. The error of the model M(G) of Pin and Baranyi (1998) was very high (60%) in the new environmental conditions studied in this paper because they were outside the interpolation region of the old model. This is not unusual with empirical models, even if the new additional region is small and close to the original one. Hence, a new model, Me(G), was created using both the new and old sets of data. McMeekin and Ross (1996) found that the pH had no effect on the growth rate of pseudomonads in the range pH 5·3–7·8. In our work, we found a slight pH influence in the range pH 5·2–6·8, which we decided not to omit. We found the indigenous food population to grow more slowly than the controlled bacteria inoculated into sterile food. All the strains used in our work were isolated from food. Hence, it would not be reasonable to assume major differences in intrinsic features of the organisms; rather, the growth differences can be attributed to the bacterial composition and, as a consequence, to the interactions within the population. In our mixed population study (Pin and Baranyi 1998), the growth of Pseudomonas spp. is not affected by other selected groups of bacteria; however, the bacteria indigenous to food may be more vigorous and competitive. Cox and Mac Rae (1988) observed that the growth of some psychrotrophic strains of Pseudomonas spp. inoculated into raw milk was slower than when inoculated in UHT milk. They explained this fact by the competitive effect of the natural population upon the inoculated strain. According to some published data, our model can also underpredict the growth rate of pseudomonads in some situations. Examples for this are the data of Cox and Mac Rae (1988), in inoculated UHT goats’ milk, and those of Gill and Newton (1977), in inoculated aseptically obtained meat slices. With these data, the percentage bias indicators were negative, about −25% and −11%, respectively. The model Me(G), on the other hand, gives faster rates (overpredictions) than the observations from UHT cows’ milk of Shelley et al. (1986), showing a positive percentage bias of about 25%. The estimation of the overall error of the model Me(G) with respect to our data in pasteurised milk was about 60%. Because all the predictions were overestimations, the percentage bias is a little smaller but not far from this value. Me(G) also overpredicts the data of Griffiths et al. (1987),


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obtained in raw milk, having a percentage bias of 33%. In pasteurised milk, Urbach and Milne (1987) observed much slower growth rates than our model predictions (139% bias). This spoilage rate is too slow to be attributable to the growth of pseudomonads. Temperature of storage and post-pasteurization contamination are the two factors with the greatest influence on the shelf-life of pasteurised milk (Griffiths 1994). Muir (1996) detected a much slower rate of spoilage in aseptically collected samples after pasteurisation than in retail products. Models for pseudomonads cannot necessarily be expected to give good predictions for growth of other postpasteurisation contaminants or for heat-resistant bacteria. All the comparisons above can be followed in Fig. 4. Comparing predictions of the model Me(G) with data of Gustavsson and Borch (1993; beef carcasses) and Drosinos and Board (1996; minced lamb) we found a smaller overall error (8%) than that calculated with our data (45%). We have presented a quantitative study on the main sources of the bias and error of a predictive model when applied to food products. We found that, when models based on broth data are validated, a difference should be made between observations obtained in sterile food inoculated with controlled strains and those obtained from naturally spoiling food, because the first scenario is closer to the conditions under which the model was generated. Primary error can be minimized by decreasing the variance of the data with greater precision in laboratory technique and/or by using more adequate models to increase the goodness of fit. Overall error has partly the same causes as does primary error, plus the increased complexity of the food environment. By means of the classification of errors described in this paper we hoped to provide a useful methodology to make the analysis and validation of predictive models more exact and accurate. ACKNOWLEDGEMENTS

The data which were generated in this laboratory were prepared by Esther Sanjuan Velazquez, Gyopar Sipos, Simone Heinz and Marialena Kanellopoulou and their work is gratefully acknowledged. Other data were received with thanks from Karen Neumeyer, University of Tasmania. J. Baranyi thankfully acknowledges the support of the UK Ministry of Agriculture, Food and Fisheries project FS2524. REFERENCES Abdullah, B., Gardini, F., Paparella, A. and Guerzoni, M.E. (1994) Growth modelling of the predominant microbial groups in hamburgers in relation to the modulation of atmosphere condition, storage temperature and diameter of meat particle. Meat Science 38, 511–526. Baranyi, J., Pin, C. and Ross, T. (1999) Validating and comparing

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