In PS and FS, the dependency of the maximum specific growth rates of LAB on storage temperature was successfully described by a square root model, showing R2 values of 0.987 and 0.969, respectively (Figure 2-3), as previously documented (Manios et al., 2009). Conversely, the growth of LAB in ES varied a lot with the type of eggplant (canned or fresh) used as ingredient (Table 2-2), with ES made of canned vegetables not supporting growth of LAB and ES made of fresh eggplant allowing growth of LAB only at high temperatures. Thus, the dependency of LAB κmax on temperature could
not be adequately described. For the validation process of the product- specific models, the work-to-be-done parameter (ho; Baranyi and Roberts,
1994) was used to account for the lag time completion under dynamic temperature conditions. The ho values estimated for each spread were
independent of temperature and product, as determined by t-test (p>0.05). Thus, an average value of 1.25 ± 0.25 and 1.64 ± 0.35 for PS and FS, respectively, was used. The predicted populations of LAB showed high correlation with the populations that grew in the examined products during storage in 4 household refrigerators (Figure 2-4; Figure 2-5). In general, the models performed well with slight under-prediction of the LAB growth in FS. Based on the facts that: (i) the growth of LAB was mainly affected by the product pH and storage temperature and (ii) the diversity of LAB species was independent of product and storage temperature, two ―unified‖ (generic) models, based on polynomial and Ratkowsky equations were developed for
initial pH, the concentration of undissociated acetic acid and storage temperature (Table 2-5). Other parameters that may influence the microbial behaviour in the spreads such as the water activity (aw) and lactate or
benzoic acid concentration could also be incorporated as a variable of the unified models. However, these factors did not change significantly among the different products of this category and therefore, it was assumed that their effect was similar in all spreads.
Similarly to the validation process of the product-specific models, the ho
parameter was calculated for all products that were used for fitting the unified models. The linear correlation (p<0.05) of lag times with the reciprocal of κmax
of LAB, suggested that ho is independent of the aforementioned factors
controlling microbial growth. Thus, the simulation of LAB growth by the unified models under dynamic chill-chain conditions were based on the average ho value of all products which equaled to 1.86 ± 0.76. Although both
models adequately described the trend of LAB growth in the different acetic acid-based products tested, the performance indices of polynomial equation were higher than those of Ratkowsky equation and so was the visible agreement of model predictions with data (Figure 2-4; Figure 2-5).
Similarly, both models were able to accurately predict the growth of LAB in feta cheese-spread (Manios et al., 2009) without the addition of the spicy green pepper, with the polynomial model, however, showing lower Bf and Af
compared to the Ratkowsky model. In contrast, both models significantly over-predicted the growth of LAB in the feta cheese spread with the spicy pepper, while the corresponding product-specific model showed overall better performance (Table 2-6). This may attributed to the antimicrobial effect of the capsaicin (Cowan, 1999) present in the spicy green pepper, which may suppress the growth of the SSOs. The potential variance in LAB growth introduced by the inhibitory effect of capsaicin in the above dairy spread ecosystem requires further experimentation in order to be accounted for by the ―unified‖ model. Polynomial equations may fit a variety of datasets due to its flexibility. However, in imbalanced datasets, which may be comprised of
areas with limited data next to areas with abundant data, the risk of over- fitting is high (Geeraerd et al., 2004; Diels et al., 2007). Thus, despite the better performance of the polynomial over Ratkowsky equation, the added value of the latter relies on its biological meaningful parameters and the consistent description of the dependency of κmax on the predictor variables,
avoiding over-fitting.
Table 2-5 Significant (p<0.05) parameter estimates (± 95 % confidence intervals) and goodness-of-fit criteria of polynomial and Ratkowsky equations fitted as two alternative “unified” models to collective data (log CFU/g) for the growth of LAB in various acetic acid-based spreads.
Equation Parameters Value -95% CI +95% CI
Polynomial Intercept 2.035 1.260 2.810 T 0.818 0.549 1.086 UAC -6.917 -8.929 -4.905 T2 0.0009 0.0002 0.0016 UAC2 0.358 0.214 0.503 T x pH -0.196 -0.261 -0.131 pH x UAC 1.259 0.874 1.644 R2 0.893 F 0.130 RMSEa 0.075 Ratkowsky Intercept 0.046 -2.371 1.108 Tmin -5.260 0.021 0.033 UACmin 7.590 -0.054 0.030 pHmin 3.793 -0.249 0.603 R2 0.800 F 0.110 RMSEa 0.093
Over-fitting was the primary reason for the disagreement between the polynomial equation-based growth simulations and the actual growth of LAB in other products of this category, such as Russian salad or beetroot spread, as opposed to the good agreement of Ratkowsky equation-based simulations (results no shown).
The description and quantification of microbial growth dynamics in different environments and foods via mathematical equations has been widely studied in the literature. However, except for the CMSICEE model and a model developed by Vermeulen et al. (2007), which are growth probability models, there are limited models for the shelf-life determination of emulsified RTE spreads or dressings of low pH. Manios et al. (2009) primarily investigated the potential of a simple square root model to predict the growth of LAB in two cheese-based spreads under real chill chain conditions. Similarly to the present study, the predictions of the product-specific models of that study were in good agreement with the observed populations of LAB during
Figure 2-3 Linear regression of the maximum specific growth rates of LAB and storage temperatures in pepper-based (●) and fava beans-based spread (■).
distribution and household storage. Although in both studies the development of such models requires extensive microbiological research and thus, reduces their usefulness by food industries, they constitute the basis for the development of mathematical models for broad food categories. In particular, starting from basic models, which study the effect of individual factors on the microbial growth, the new trend of predictive models is to gradually incorporate as much influencing factors as possible (Bernaerts et al., 2004). In this way, the accuracy of the predictions of the model increases, while the necessity of time-consuming analysis decreases. Despite the proper
Table 2-6 Bias (Bf) and accuracy (Af) factors of the models predicting the
growth of LAB in pepper spread and fava beans spread, after exposure to fluctuating time-temperature profiles
Product Model Time- temperature profiles Bf Af Feta cheese- based spread Square root model A 1.09 1.09 B 1.16 1.16 Polynomial model A 1.01 1.02 B 1.11 1.11 Ratkowsky model A 1.06 1.06 B 1.25 1.25 Feta cheese- based spread with spicy green pepper Square root model A 1.03 1.05 B 0.85 1.17 Polynomial model A 1.44 1.44 B 1.52 1.52 Ratkowsky model A 1.38 1.38 B 1.47 1.47
temperature configurations (4oC) set by the consumers in household refrigerators, the actual storage temperature in the validation trials of the present study ranged from -1.2 to 13.9oC, which include values outside the temperature interpolation area of the models. Likewise, handling of some products during validation experiments also resulted in short exposure to temperatures up to 33.7oC (Figure 2-4; Figure 2-5). These abrupt fluctuations could induce an additional lag time. The high performance of the ―unified‖ models despite these issues addresses the robustness of the present generic approach.
Figure 2-4 Growth of lactic acid bacteria in pepper spread under four (A-D) time-temperature profiles (----) in four household refrigerators (A-D), as predicted by the polynomial model (▬), the Ratkowski model (▬), or the product-specific model (- - -) in conjunction with the observed populations (●).
Figure 2-5 Growth of lactic acid bacteria in fava beans spread under four (A-D) time-temperature profiles (----) in four household refrigerators (A-D), as predicted by the polynomial model (▬), the Ratkowsky model (▬), or the product-specific model (---) in conjunction with the observed populations (●).