Weibull Distribution Function Based on an Empirical Mathematical Model for Inactivation of Escherichia coli by Pulsed Electric Fields

Transcription

1 1007 Journal of Food Protection, Vol. 66, No. 6, 2003, Pages Copyright q, International Association for Food Protection Weibull Distribution Function Based on an Empirical Mathematical Model for Inactivation of Escherichia coli by Pulsed Electric Fields D. RODRIGO, 1 G. V. BARBOSA-CÁNOVAS, 2 A. MARTÍNEZ, 1 AND M. RODRIGO 1 * 1 Instituto de Agroqu mica y Tecnolog a de Alimentos, Consejo Superior de Investigaciones Cient cas, Apartado de Correos 73, Burjassot, Valencia, Spain; and 2 Department of Biological System Engineering, Washington State University, Pullman, Washington , USA MS : Received 24 July 2002/Accepted 18 October 2002 ABSTRACT The pulsed electric eld inactivation kinetics of Escherichia coli suspended in orange juices with three different concentrations of carrot juice (0, 20, and 60%) was studied. Electric eld strengths ranged from 25 to kv/cm, and treatment times ranged from to 3 ms. Experimental data were tted to Bigelow, Hülsheger, and Weibull distribution functions, and the Weibull function provided the best t (with the lowest mean square error). The dependency of each model s kinetic constant on electric eld strength and carrot juice concentration was studied. A secondary model was developed to describe the relationship of Weibull parameters a and n to electric eld strength and carrot juice concentration. An empirical mathematical model based on the Weibull distribution function, relating the natural logarithm of the survival fraction to treatment time, electric eld strength, and carrot juice concentration, was developed. Parameters were estimated by a nonlinear regression. The results of this study indicate that the error rate for the model s predictions was 6.5% and that the model was suitable for describing E. coli inactivation. New types of refrigerated juices, prepared by mixing fruits and vegetables, are being marketed to satisfy consumer demand for natural, high-quality, healthy products. These new juices contain functional components that improve consumer health. One of these juices, currently being manufactured in various countries, combines the excellent sensory and nutritious characteristics of orange juice with the functional properties of carrot juice, providing a substantial contribution to consumer health. However, when these juices are thermally processed to extend their shelf lives, changes take place in their color, aroma, and avor, as well as in many of the functional characteristics that are important for these kinds of foods (10). Consequently, producers have become interested in nding alternatives to thermal pasteurization (6). Pulsed electric eld (PEF) treatment is one of the nonthermal technologies being studied intensively to evaluate its potential as a process alternative or complement to thermal pasteurization. Nevertheless, food should be not only nutritive but also microbiologically safe. New technologies must demonstrate suf cient microbial inactivation of pathogen microorganisms. From a safety standpoint, Escherichia coli contamination of orange juice is of the greatest concern. For an outbreak that occurred in India in 1992 (1), the number of cases is unknown, as is the source of contamination; however, the probable causes were poor sanitation practices and a bad facility design (13). Many studies have demonstrated that PEF treatment has lethal effects on E. coli and that the * Author for correspondence. Tel: ;Fax: ; mrodrigo@iata.csic.es. rate of inactivation increases as treatment time and electric eld strength increase (4, 16). A 5.6- to 4.2-log reduction of E. coli O157:H7 was obtained with a PEF treatment combined with organic (benzoic and sorbic) acids (12). However, data on PEF kinetic inactivation and on the effect of environmental factors and kinetics parameters are lacking. The basic models used for interpreting survival curves focus on rst-order relationships. However, as kinetic data on PEF inactivation of microorganisms accumulate, more and more cases are encountered in which there is no clear evidence that PEF survival curves can be interpreted with a rst-order model. Therefore, there is a need to develop microbial inactivation models involving PEF that are focused on the microbial population of interest for food safety and to develop and evaluate the corresponding kinetic models (8). In the eld of thermal inactivation, various empirical models relating microbial inactivation to environmental factors such as ph or the concentration of a chemical compound (e.g., NaCl) have been developed (5, 11, 15). One of the most important environmental factors affecting the ef ciency of PEF treatment is conductivity, and there is no model relating PEF kinetic parameters to electric eld strength and conductivity. In the present work, kinetic studies were carried out to describe the nature of the inactivation curves of E. coli as affected by electric eld strength and conductivity, which was modi ed by varying the orange juice/carrot juice ratio of the treatment medium. The kinetic data were used to develop an empirical mathematical model that appropriately relates the surviving fraction of E. coli to treatment time,

2 1008 RODRIGO ET AL. J. Food Prot., Vol. 66, No. 6 FIGURE 1. Survival curves for E. coli cells at kv/cm and for carrot juice concentrations of 0% (l), 20% (m), and 60% ( ). electric eld strength, and the percentage of carrot juice present in the juice mixture. MATERIALS AND METHODS Culture preparation. A freeze-dried pure culture of Escherichia coli ATCC 8739 was provided by the Spanish Type Culture Collection. The culture was rehydrated with 10 ml of nutrient broth (NB; Scharlab Chemie S.A., Barcelona, Spain). After 20 min, 5 ml of the culture was inoculated in 100 ml of NB and incubated at 8C with continuous agitation at 200 rpm for 2 h. Forty milliliters of the resulting culture was transferred into 0 ml of NB and incubated for 8 h under the same temperature and agitation conditions to obtain cells in an exponential growth stage. The cells were centrifuged twice at 4,000 3 g at 48C for 15 min and then resuspended in NB. After the second centrifugation, the cells were resuspended in 15 ml of NB with 20% glycerol and then dispensed into 2-ml vials. The 2-ml samples were immediately frozen in liquid nitrogen and stored at 2808C until they were needed for the kinetic inactivation studies. Treatment medium and inoculation. Pasteurized mixed juices with different orange juice/carrot juice ratios (100:0, 80:20, and :60) were prepared from pure orange juice and pure carrot juice and stored at 28C pending PEF treatment. The electrical conductivity values for these mixtures at 258C were 0., 0.46, and 0.65 S/m, respectively. Conductivity measurement, based on the principle of the Wheatstone bridge, was carried out with a Crison 525 conductimeter (Crison Instruments S.A., Barcelona, Spain). Particle size was adjusted to 0.29 mm. One liter of juice was inoculated with a previously thawed 2-ml vial of the frozen microorganism with a concentration of CFU/ml. PEF treatment system. An OSU-4D bench-scale continuous PEF system, designed at Ohio State University, was used to treat the samples. Six co- eld treatment chambers (diameter, 0.23 cm; gap distance, cm) were connected in series. Two cooling coils were connected before and after each pair of chambers and submerged in a circulating refrigerated bath to maintain a treatment temperature in the designated range (15 to 8C) (inlet temperature, 158C; maximum treatment temperature, 8C). Pulse waveform, voltage, and intensity in the treatment chambers were recorded with a digital oscilloscope (Tektronix TDS 210, Tektronix Inc., Wilsonville, Oreg.). The ow rate was set at 60 ml/min with a peristaltic pump (Millipore Corporation, Bedford, Mass). A square-wave bipolar pulse duration of 2.5 ms was used. Treatment times ranged from to 3 ms, and the electric eld strengths were set at 25,,, and kv/cm. Samples were collected after each treatment time and were serially diluted in sterile 0.1% buffered peptone water, plated on nutrient agar, and incubated for 24 h at 8C. The experiments were carried out in duplicate. Mathematical models. Three mathematical models were used in this study. The Bigelow (2) model is as follows: t log(s) 5 2 (1) D where S is the survival fraction at treatment time t, and D is the decimal reduction time (mathematically, the inverse of the inactivation curve slope). The Hülsheger (9) model is as follows: ln(s) 5 2b(ln t 2 ln t C ) (2) where b is the regression coef cient, t is the treatment time, and t C is the most critical treatment time obtained (the longest treatment time for which the survival fraction is 1). The Weibull distribution function (20) is as follows: 1 2 n t ln(s) 5 2 (3) a where a and n are scale and shape factors, respectively; the n factor describes the shape of the survival curve so that when n, 1 the survival curve is concave (it forms tails), when n. 1 the survival curve is convex (it forms shoulders), and when n 5 1 the survival curve is a straight line on a ln scale (equal to the Bigelow model). Parameter estimation and global model building were accomplished by nonlinear regression with Statgraphics Plus 5.0 software. To assess predictions made by the global model, an accuracy factor parameter (Af) (18) was used: [ S z log(predicted/observed)z Af 5 10 ]/n (4) where n is the number of observations used to make the calculations. The predicted/observed ratio refers to the relationship between the survival fraction predicted by the model and the one obtained experimentally.

3 J. Food Prot., Vol. 66, No. 6 PEF INACTIVATION MODEL FOR E. COLI 1009 TABLE 1. Bigelow model parameters obtained by tting experimental data to equation 1 a TABLE 2. Hülsheger model parameters obtained by tting experimental data to equation 2 a % carrot juice E (kv/cm) D b (min) MSE % carrot juice E (kv/cm) Ln t b C (min) b b MSE a E, electric eld strength; D, decimal reduction time; MSE, mean square error. b Mean 6 95% con dence interval. a E, electric eld strength; t C, critical treatment time; b, regression coef cient; MSE, mean square error. b Mean 6 95% con dence interval. RESULTS AND DISCUSSION The logarithm of the survival fraction was plotted against treatment time for each of the electric eld juice mixture combinations studied. Figure 1 shows examples of survival curves at kv/cm. As the concentration of carrot juice increases, the effectiveness of the treatment diminishes, as re ected in a decrease of the decimal log reductions reached (from 2.8 to 1). This tendency was exhibited for each of the electric elds studied. Another important characteristic of the survival curves obtained for all of electric elds and types of juices studied is the lack of linearity. Figure 1 shows that the survival curves tend to form tails that are more marked as the carrot percentage diminishes. Decimal logarithms and natural logarithms of the survival fractions were adjusted by nonlinear regressions (Statgraphics Plus 5.0) to the Bigelow and Hülsheger models and to a Weibull distribution function. Table 1 shows the values for the Bigelow model parameters. The kinetic parameter D (decimal reduction time) diminishes when the electric eld strength increases and when the percentage of carrot juice present in the mixed juice diminishes so that, as stated above, it diminishes when the treatment intensity increases. The goodness of t was evaluated with the mean square error (MSE) (18), which represents a measure of accuracy computed by squaring the individual error of each item in the data and then nding the average or mean value of the sum of those squares. MSE values for the Bigelow model ranged from to Table 2 shows the values for the Hülsheger model parameters. The b value is an index of the resistance of E. coli cells to treatment. Like the D-value, it diminishes when the electric eld intensity increases and when the percentage of carrot juice in the mixed juice diminishes. MSE values for the Hülsheger model ranged from 0.12 to Table 3 shows the values for the Weibull distribution function parameters. The best t (lowest MSE) was obtained when the experimental data were adjusted to the Weibull distribution function (0.002 to 0.076). As can be seen in the Table 3, the Weibull distribution has two parameters, a and n, the scale and shape factors, respectively. The scale factor, which represents a measure of the treatment resistance, diminishes when the treatment intensity (electric eld strength) increases and when the proportion of carrot juice in the mixed juice diminishes (Table 3). The variability of the shape factor n was studied by means of an analysis of variance (Fig. 2). There were no signi cant differences (P. 0.05) between n values for the different electric elds applied. Similar results were obtained by other investigators in studies on the heat inactivation of Bacillus cereus (5), the effect of high hydrostatic pressure on Bacillus subtilis (7), and the effect of an electric eld on Lactobacillus plantarum (17). However, it appears that the orange juice/carrot juice ratio in the treatment me- TABLE 3. Weibull model parameters obtained by tting experimental data to equation 3 a % E carrot juice (kv/cm) a b n b MSE a E, electric eld strength; a, scale parameter; n, shape parameter; MSE, mean square error. b Mean 6 95% con dence interval.

4 1010 RODRIGO ET AL. J. Food Prot., Vol. 66, No. 6 TABLE 4. Parameter estimates obtained by tting the global model (equation 7) to the total experimental data Parameter a b d «f g Estimate Asymptotic 95.0% con dence interval FIGURE 2. Shape parameter (n) values and their 95% con dence intervals for electric eld strengths of 25,,, of kv/cm and for carrot juice concentrations of 0% (l), 20% (m), and 60% ( ), obtained by tting the Weibull distribution function (equation 3) to experimental data. The shaded area represents the 95% con dence interval of the n mean value for each carrot juice concentration. dium affects the n value, which increased (P # 0.05) from a mean of 0.52 for pure orange juice to a mean of 0.72 for the mixture of 60% carrot juice and % orange juice. As Peleg and Cole (14) and Van Broekel (19) reported, environmental factors such as ph, ionic strength, and the concentrations of certain components seem to in uence the shape parameter n. In the present study, the value of n was always,1, indicating that the microbial population had a spectrum of PEF resistance levels and re ecting the progressive elimination of sensitive members of the population and the increased strength of the survivors (14). An analysis of variance was performed to determine the effect of the orange juice/carrot juice ratio on the kinetic parameters (D, b, and a) for the Bigelow, Hülsheger, and Weibull models, respectively. The results of the analysis show signi cant differences (P # 0.05) between the parameters values for different types of juices. The values of the parameters decrease as the carrot juice percentage decreases. Carrot juice has a very high electrical conductivity level (0.87 S/m at 258C), so an increase in the concentration of carrot juice leads to an increase in the conductivity of the mixed juice (Fig. 3). Foods with high conductivities generate smaller peak electric elds across the treatment chamber. Consequently, as Wouters and Smelt (21) reported, an increase in conductivity produces a decrease in the effectiveness of the treatment, which is re ected in a decrease in the values of the kinetic parameters. The Weibull distribution function, which was considered the most appropriate for interpreting the inactivation of E. coli, was used to develop an empirical mathematical model to explain the combined effect of electric eld strength and medium conductivity on this inactivation. The conductivity was modi ed by changing the orange juice/ carrot juice ratio. As Figure 3 shows, there is a linear relationship between carrot juice concentration and conductivity such that for practical industrial purposes the variable represented in the model is the carrot juice concentration. The relationship between parameter a, electric eld strength, and carrot juice concentration was determined by means of a multiple linear regression (Statgraphics Plus 5.0) with the natural logarithm of a as the dependent variable. Equation 5 shows the relationship obtained (resulting in a determination coef cient [r 2 ] of 0.980). FIGURE 3. Relationship between conductivityand the carrot juice concentration in the treatment medium. FIGURE 4. Survival curves for E. coli cells at kv/cm for carrot juice concentrations of 0% (l), 20% (m), and 60% ( ) tted to the Weibull distribution function (equation 3).

5 J. Food Prot., Vol. 66, No. 6 PEF INACTIVATION MODEL FOR E. COLI 1011 TABLE 5. Correlation matrix for parameter estimates obtained by tting the global model (equation 7) to the total experimental data a b d «f g a b d «f g ln(a) (% carrot juice) (% carrot juice) E E 2 (5) where E is electric eld strength. According to the previous results, parameter n is considered dependent on the carrot juice percentage (Fig. 2) in accordance with the following relationship (resulting an r 2 value of 0.999). n (% carrot juice) (6) Relationships obtained in equations 5 and 6 were introduced into the equation of the Weibull distribution function (equation 3), resulting in the following global model: ln(s) 5 2{t/{exp[a 1 b 3 (% carrot juice) 1 x 3 (% carrot juice) 2 2 d 3 (E) 1 «3 (E) 2 ]}}[f3(% carrot juice)1w] (7) To improve the estimates of the parameters, all of the data points (ln(s)) were tted to equation 7 by nonlinear regression (Statgraphics Plus 5.0). The application of this analysis improves the precision of the estimates because it avoids possible errors made through the estimation of intermediate parameters and because it uses all of the raw data to estimate the parameters (5). The parameter estimates and con dence intervals obtained are shown in Table 4. The experimental data were compared with those predicted by the model, and the goodness of t was calculated with the accuracy factor (Af), which was 1.065, indicating FIGURE 6. Response surface predicted by the global model (equation 7) for the inactivation of E. coli by PEF. an error rate of 6.5% for the predictions. Figure 4 shows the t of experimental data to those predicted by the model. Table 5 shows the correlations between the model parameters. There are only three combinations with correlations whose absolute values are.0.8. The analysis of the residuals shows that they are randomly distributed (following a normal distribution) with an average of 0 (Fig. 5). This result was con rmed by the Kolmogorov-Smirnov test. The presence of atypical residuals was observed, and their absolute values were always The results of all of these analyses con rm that the model developed is capable of describing the relationship between inactivation of E. coli by PEF, electric eld strength, and carrot juice concentration. Figure 6 shows the response surface and the relationship between the natural logarithm of the survival fraction, the treatment time, and the percentage of carrot juice in the juice mixture. The gure shows that inactivation increases as treatment time increases and the carrot juice concentration diminishes. For application purposes, a correlation between electrical conductivity, electric eld strength, treatment time, and inactivation of E. coli may help to determine the combination of variables that maximizes PEF applications. From the results presented here it can be concluded that the model developed satisfactorily relates electric eld strength, treatment time, and carrot juice concentration with the natural logarithm of the survival fraction treated by PEF, with the error rate for the model s predictions being about 6.5%. ACKNOWLEDGMENTS This work was carried out with the nancial support of EU and Spanish CICYT project 1FD C03-01.The authors thank Auri Fernández for her help in the data analysis. REFERENCES FIGURE 5. Histogram of the residuals resulting from the t to the global model (equation 7). 1. Anonymous Risk assessment: unpasteurized fruit juice/cider. Health Canada. Available at: food/juice-outbreaks.htm. 2. Bigelow, W. D The logarithmic nature of thermal death time curves. J. Infect. Dis. 29: Evrendilek, G., Q. H. Zhang, and E. R. Richter Inactivation of Escherichia coli O157:H7 and Escherichia coli 8739 in apple juice by pulsed electric elds. J. Food Prot. 62: Fernández, A., J. Collado, L. M. Cunha,, M. J. Ocio, and A. Mar-

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